Here's a great place to start: Interplanetary Cessna - XKCD What If?
As a human on Earth, you have a great advantage in determining what you need, in that the Sol system has 4 gas giants of its own, each fairly well-studied and visited by numerous probes. Randall Munroe's What If question focused on using an ordinary Earth airplane, the Cessna 172 (one of the most common in North America) which, for most of the gas giants, turns out to be an example of what not to do.
Let's model your own gas giant after Jupiter; it's not quite as friendly as Saturn to air travel, but it's the first gas giant a casual reader will think of, and most planets we know of are "super-Jovian" gas giants even bigger than Jupiter. The problems you'll face get bigger as the planet does, because the planet's gravity limits the thickness of an actual gaseous atmosphere. Your habitable range of Earth-like atmospheric pressures will end up about the same height above the transition to a liquid layer, meanwhile the gravity of that ever-bigger ball of liquid hydrogen under you will make it that much harder to stay afloat as the size of the planet increases. If you can solve Jupiter's problems, it will be that much easier to solve problems of a smaller giant like Saturn, and you'll have that much more of a leg up on the problems of a superjovian.
Here's a short list of the things your Jovian airplane will need:
Oxidizer where fuel normally goes. Most gas giants are primarily hydrogen; there's relatively little of anything else but hydrogen in any one place to make a planet of this size (while we have found "mega-Earths", primarily solid planets many times Earth's mass, superjovians are much more common among cataloged exoplanets to date). That means that your airplane is flying in its own fuel, unlike an Earth aircraft which flies in its own oxidizer. Your engines, if "air-breathing", will actually aspirate the hydrogen atmosphere, combine it with oxygen from its internal storage tanks, and ignite that mixture to produce thrust from heated gas. It's kind of a backwards way to think about it, but as long as you get the mixtures right it works exactly the same way (other than the fact that mostly for safety reasons but also for some practical ones, modern airliners don't run on hydrogen).
Big wings and lotsa thrust. At atmospheric pressures comparable to Earth's atmosphere, the gravity of Jupiter is about two and a half times as much as on Earth. Go higher to reduce relative gravity, and you'll have less gas flowing over your wings to generate lift and through your engines to produce thrust (and you won't get anywhere near Earth gravity while still in anything you'd call Jupiter's atmosphere). So, you need a plane with about 2.5x the lift-to-mass and thrust to mass ratio of a plane designed for Earth, because that same plane, of the same mass, will weigh 2.5x as much in Jupiter's atmosphere, and so lift-to-weight and thrust-to-weight will be reduced by that same factor.
We have to deal with this kind of math fairly regularly in designing Earth aircraft. The interior volume of a passenger aircraft is a rough analog of the plane's max takeoff weight, due to a combination of regulations, physics, and plain old common sense about things like minimum airspace per passenger that keeps them coming back for another flight in your sardine cans. The heavier the aircraft, the larger its wings must be, however wings don't scale the same way fuselages do; if the wing's cross-section gets too thick, you end up increasing leading-edge drag too much, so the most efficient airfoils for large aircraft remain relatively thin as wingspan and chord length increase. This means that superjumbos like the B747 and A380 have much larger wing areas as a ratio of their fuselage "footprint" compared to smaller "single-aisle" jets like the B737 and A320.
Your Jovian aircraft, for the same volume, will weigh 2.5x as much as on Earth. We can compensate for that in Earth's atmosphere by simply increasing mass by 2.5x for the same volume of aircraft. As it happens, the Boeing 737-800 and the Boeing 767-300ER have almost exactly the necessary relation in max take-off weight. So, in theory, if you take the 767's wings and engines and put them on a 737, you have about the necessary lift it would take to fly that 737 on Jupiter. That would increase the wingspan by about two-thirds (from 93' to 156'), and would about triple wing area (979.9 ft^2 to 3050 ft^2). You'd also go beyond any possibility of the engines being underslung on the wings (the up-engine to the -300 series already necessitating the noncircular "hamster-pouch" engine nacelles) and so you would have to find some other agreeable place on the 737 to put the 767's 8-foot-diameter CF6 engines.
- Totally sealed interior, at least for oxygen-breathing inhabitants. Jupiter's atmosphere again being mostly hydrogen, you really don't want any of that within livable spaces of the aircraft. Hydrogen has one of the wider flammability ranges of any flammable gas; in concentrations as low as 5% and as high as 95% in otherwise Earth-proportioned air mixture, you will get ignition, and in anything but the absolute extremes of that range it will explode quite spectacularly, with the stoichiometric ratio of hydrogen in air being about 9.5%. So, while your plane doesn't actually have to maintain that much of a pressure differential to fly in Jupiter's atmosphere (though you would get a similar drag-reducing benefit flying higher than Earth sea level pressures, and you'd probably want some sort of crush pressure rating in case a plane descended deeper for any reason), the passenger cabin must be totally airtight.
The problem is that airliners are not and have never been airtight; they circulate outside air into the cabin from the initial compression stages of their engine, cooled with heat exchangers, to maintain sea level pressures within the cabin. Even at a quarter-billion dollars or more apiece, it's just too expensive for the manufacturer to deliver an airplane that is totally airtight, and too expensive for the airline to keep it that way, not to mention that changing the air within the cabin is the easiest way to mitigate objectionable smells (like the passenger three rows up who really should have showered before getting on a plane). Positive cabin pressure, fairly easy for most phases of flight, would ensure that any hydrogen-oxygen mixture due to leakage happens outside the plane, but that oxygen would need to be replaced, which is its own problem.
- Total IFR with Satellite navigation. We casually view a city built over a gas giant to look like Bespin from Star Wars; a towering metropolis nestled conveniently in the upper cloud layers of the planet from which it's extracting useful material.
In reality, at Earth pressure, what you're going to get looks less like Bespin and more like Beijing, when the Olympics aren't in town:
In flight, there are two basic sets of rules for flying. Visual flight rules or VFR are for when you can see where you're going. Instrument flight rules or IFR are for when you can't. Jupiter's atmosphere at Earth-like pressures will definitely be the latter, requiring pilots to rely primarily on instruments to navigate. That in itself is not as big a problem as it sounds; airline pilots fly using IFR almost all the time for a combination of safety and practicality. The problems center around the very basic difference between Earth and Jupiter; nothing is anchored to the ground, because any "ground" to speak of is tens of thousands of miles deeper and thousands of degrees hotter than anyone or anything can survive.
Jupiter does have a magnetosphere; quite a strong one in fact, about ten times the field strength and 18000 times the magnetic moment, so the biggest problem you'll have with a traditional compass is dealing with severe magnetic dip (the magnet will orient along the lines of the magnetic field, which are not perpendicular to gravity or parallel to the "surface" along which you're flying). A fluid-filled spherical compass would need to be a little larger than usual to keep the azimuth lines accurate and readable at higher latitudes of the sphere as the compass dips, or you could just ditch the magnet and use solid-state magnetometers, which are what modern airliners use (because the data can be easily fed into digital flight displays and computerized flight directors/autopilots).
The bigger problem is that, very unlike any terrain feature on Earth which would only move relative to very far-away objects on other plates, and even then only at a rate of a couple inches a year, a city floating along in Jupiter's atmosphere will be carried along by zonal jets averaging about 60mph, and buffeted by true storms that exceed even the Enhanced Fujita scale for tornadoes, with wind speeds over 350mph. These zonal jets move in opposing directions, so more than one city at more than one latitude (which you'd have, otherwise why bother flying) would mean that flight plans between the two cities would look more like orbital trajectories than terrestrial travel plans. Your reckoned course from one city to the other would have you flying along a path that compensates not only for the distance the destination city will move along its own zonal stream during your time of flight, but the relative wind speeds of all the layers you'll travel through on your way there, which will blow you left and right as you move north or south around the planet. You do have to account for the prevailing wind direction and average speed on Earth, as you will be flying through a mass of air that is itself moving relative to Earth's surface, but the triangulation required for your average terrestrial flight plan would have nothing on the math to reckon your course around Jupiter.
And, weather conditions can change. "Dead reckoning", a purely mathematical, instrument-based course-calculation, is called that for a very good reason; you are right, or you're dead. Even IFR on Earth requires the pilot to use visual cues to correct for any inaccuracy, and most private civilian navigation is done using waypoints based either on terrain features or navigational beacons at precisely-known locations (which broadcast signals that allow you to know your exact bearing from that beacon, and knowing the angle of azimuth from each beacon gives you a triangular fix of your location). Once again, nothing on Jupiter's surface is in a fixed location, so there is no such thing as a "terrain feature" or a "navigational beacon" from which to triangulate.
The solution required would be similar to the system we've had since about 1997; the Global Positioning System. A constellation of satellites in a very precisely-calculated system of orbits, with very finely-synchronized chronometers aboard, such that every position on earth is within line of sight of at least 4 of these satellites. The position of any satellite at any given point in time after the system's initial epoch can be calculated based on their known, stable orbits, and each satellite is sending a time-coded bitstream that allows the receiver to calculate the "time of flight" of that signal, which when multiplied by the speed of light, gives you the distance to that satellite. Knowing the distance to one satellite, along with its current position over the earth, means you are somewhere along a near-circle on the idealized Earth's surface (a perfectly smooth oblate spheroid) that you'd get by drawing on a globe using a pencil of that length anchored at the position of the satellite. The measurement from two satellites gives you a second intersecting circle that ideally provides a maximum of two possible locations on the Earth's surface that you could be, and three different satellites will in most cases give you a unique position fix within the margin of error of the distance measurements. Four are typically used in GPS to guarantee an accurate fix, because the orbital dynamics of the constellation mean that two satellites can be relatively close in position which reduces accuracy.
Navigation in Jupiter's atmosphere, within which nothing is in a totally fixed position, would require a "JPS" system that would give similar distance-based position fixes within Jupiter's atmosphere. Ideally, signals could be received from below what would be Earth's horizon, which would allow altitude calculations based on the calculation of your position in 3D space, reducing reliance on a pressure altimeter (vital but troublesome even on Earth, as surface barometric pressure can change by enough to matter in less than an hour). If you're going to get from one city to the other, and you're not absolutely sure of the wind speeds in the zonal jets between the two cities, this is the only way you'll get close enough.
- A steady supply of both oxygen and an inert gas. When we're talking about breathing oxygen within a hydrogen atmosphere, things get much more difficult. The pressure differential isn't even that much of an advantage, because you still can't have any leaks; any slight pressure differential between inside and outside will leak one gas into the other, and as I said before, hydrogen gas in an oxygen environment has a ridiculously wide window of flammable concentration, which only gets wider when you enrich the atmosphere with a higher partial pressure of oxygen.
That's the first problem; Apollo, the Space Shuttle and even early space stations like Skylabs and Mir got by on a low-pressure, oxygen-rich atmosphere. This reduces the required pressure handling of the spacecraft, and also reduces the amount of tanks of gas that have to be dealt with to maintain the proper atmosphere. The ISS, designed for longer-term stays (and with more frequent unmanned resupply launches) incorporates nitrogen into the mix, which reduces flammability in the environment, increasing safety, and also reduces the aging effects of high-oxygen environments on human tissues ("antioxidants" are essential vitamins for a real good reason; while we run on oxygen, it also causes cellular damage and aging, and these effects increase with concentration). Either nitrogen or argon (a totally inert gas) would be that much more necessary when the outside environment isn't a vacuum, but instead a ready supply of go-boom. Airlocks can't just pressurize with an atmospheric mix, they have to first purge the hydrogen completely before adding oxygen, otherwise you're creating a fuel-air bomb in your airlock as you oxygenate it. That means nitrogen is a "consumable"; you blast it into your airlocks as an intermediate step between introducing oxygen or hydrogen into that chamber, so those two gases are never in sufficient partial pressures to become a fire hazard. This will almost certainly require expelling quite a bit of nitrogen out into the outer atmosphere, just to be sure you have all the hydrogen out.
Overall, a slightly positive air pressure within oxygenated habitats is preferable, because then any leaks at least leak oxygen out into the surrounding air where wind currents dissipate it into harmless concentrations relatively quickly. But this creates a problem; that oxygen leakage is gone forever. You can't reclaim it from the Jovian atmosphere. Neither, for that matter, can you reclaim the nitrogen. Once it's been released into Jupiter's atmosphere, it has to be replaced from some other source.
Why is that, you ask? Well, because these gases are denser. At the same pressure, both oxygen and nitrogen weigh much more than hydrogen, which is why a balloon filled with it floats in Earth's air. In Jupiter's hydrogen atmosphere, the opposite happens; any oxygen that enters the Jovian atmosphere will sink into the depths of the planet, until it becomes hot enough that combustion to produce water vapor (or, for nitrogen, hydrogenation to produce ammonia) becomes favorable. That water is even denser, and will continue to sink into the increasingly dense, high-pressure supercritical liquid mostly-hydrogen oceans of the Jovian "surface", far deeper than any human man or machine could hope to venture. Even if we could get down into deeper layers, the best we can hope to pull back is ammonia (NH3) which is theorized to exist in some abundance in the intermediate layers of Jupiter's atmosphere. We can dehydrogenate ammonia pretty easily by burning it in oxygen (that process is far easier than making ammonia on Earth; the Haber process requires temperatures and pressures not unlike what you'll find a few miles below Earth pressures on Jupiter), giving us some nitrogen gas back (we can reclaim the oxygen from the water with electrolysis or high-temperature cracking, but neither are an efficient process).
Any oxygen release into the Jovian atmosphere is likely a total waste; the resulting water will be denser at any ambient pressure than anything else around it until the supercritical steam reaches the solid core of the planet, composed of everything Jupiter's Hoovered up over the eons. And you're 630 million miles from the closest breathable natural atmosphere in the Solar System, at the very closest; when Earth and Jupiter are on opposite sides of the sun, you're a couple billion miles from anything readily breathable.
There is some good news; Jupiter's moon Europa is an ice ball, completely covered in water ice (very likely with a liquid water ocean under that), and it's the second-closest major moon in orbit to Jupiter, so it's a relative hop-skip-jump out of the upper Jovian atmosphere compared to getting anywhere else. It would, if we're talking colonization, be a much easier "errand trip" to send ships to collect and bring back some of Europa's water, which can then be drunk, used in hydroponics, electrolyzed to oxygen, whatever use the Jovian colonies have for it. Europa also has a thin oxygen atmosphere, caused by radiolysis of the water molecules of the ice surface by a combination of solar radiation and charged particles from Jupiter's magnetosphere. Compressing that to any useful pressure would likely be a waste of time during a manned mission - the atmosphere is about 1 trillionth the pressure of Earth sea level - but it could be a worthwhile endeavor to land solar-powered compressors that could work for months on end between trips to produce relatively easy-to-use bottles of compressed or liquid oxygen.
