I'm looking for a pseudoscience, hand waving answer. I don't need fast travel to other stars, I just need something that can zip around within solar systems (even ones larger than our own). This would be something wealthy or even upper middle class people could do, not just for astronauts. For instance, take a 6 hour tour that passes by some features of Jupiter, and then some of its moons. I don't want huge fuel tanks that need to be filled constantly. A couple different kinds of propulsion would be an added bonus. I assume this is hand waving, even in a technologically advanced civilization. I may just gloss over it, but I was curious what people would suggest.
As @Andrew Dodds commented, you can have any propulsion you want (really, it's up to you), you are still limited by acceleration tolerance and speed of light.
Let's take your 6-hour trip to Jupiter for example. Spoiler : it's not going to happen. We'll consider your travelers can withstand for several hours an acceleration of 3Gs. 3Gs for hours is really uncomfortable, but survivable (unless you have some health condition). So, we're talking of an acceleration of about 30m/s². In 3 hours, this means we reach a speed of 324km/s. Why 3 hours ? Well, because now we have to decelerate in order to do a fly-by and come back ! So, your 6-hours trip will be at a mean speed of 162km/s. That's a lot, but still less than 0.05% of the speed of light. How much distance can we cover ? About 3,500,000 km.
Sadly, at its closest to the Earth, Jupiter is 4 AU away. Or 600,000,000 km.
How much time will it take with the 3Gs limit ? A bit less than 80 hours. More than 3 days. You maximum velocity would be around 4200km/s (a bit more than 1% of the speed of light), which is quite a dangerous speed : any tiny speck of dust could do enormous damages : 1 milligram of matter hitting you will be roughly equivalent to the explosion of 2 kilograms of TNT.
Why am I pointing this out ? Well, because if you want solar systems traveling within hours, your engine will have to handwave the Gs limit, by suppressing the effects of a strong acceleration. In fact, your ship should have some "inertia suppressor". And if your ship can do that, propulsion is not really a problem : if you can manipulate inertia (rather magically, like any sufficiently advanced technology), you can achieve great acceleration without the need for much propulsion. Just imagine that your inertia suppressor device makes your ship behave like it had a mass of 1kg. You won't need much fuel to hurl it at great speed. Just add some defense system against the tiny but lethal specks of dust that your ship could encounter.
Also, keep in mind that the last limit, speed of light, will let you cover a distance of 25,920,000,000 km maximum in one day (with a negligible acceleration and deceleration time), or 173 AU, about the double of the diameter or Pluto's orbit. Larger stellar systems (and ours is in the small range) may be hard to navigate in a matter of hours.
If you want to reach anywhere in the solar system using a continuous acceleration propulsion system and you limit the rate of acceleration to one gravity (1 g) this will limit you to travelling to the Moon. Approximately four hours at a constant acceleration of 1 g. The 1 g is essential if you want to travel in comfort, and everybody lives with a constant 1 g due to gravitation.
Assuming you want to travel somewhere relatively close to good old Mother Earth, say, like the planet Mars. Now in Philip K Dick's novel Dr Futurity the protagonist is deported to Mars in a relatively small capsule-type of vehicle. The entire trip took one hour. This places it in the ball park of what we are looking for.
Now the distance of the planet Mars varies according to its location on its orbit and the position of Earth on its own orbit. At its closest Mars is roughly three and half light minutes distant and at its furthest, when Mars is on the opposite side of the Sun, it is twenty light minutes away.
Therefore, if the Dr Futurity trip was to Mars at its closest, then its average velocity will be 0.05833 c or 17,500 km/sec. While if it was heading to Mars at its furthest distance the average velocity will be one-third of lightspeed or 100,000 km/sec. In each case, the space capsule will take fifteen minutes acceleration time to reach its average velocity. (Please note in both cases, this assumes the space capsule accelerates and decelerates along the entire path of its journey to Mars, so it is constantly accelerating.) The constant accelerations are, for the shortest trip 1984.13 g and for the longest trip 11,337.87 g.
Ultrahigh rates of acceleration like these will kill unprotected human beings. Therefore, the form of propulsion either needs to be such that human beings will be unaffected by the ultrahigh acceleration or there is a protection mechanism to nullify any adverse effects. For the latter protection mechanism, one of the simplest ways of doing so is hand wave a counter-acceleration system into existence. Assuming it does what it says on the tin, human beings travelling in ultrahigh acceleration craft will be unaffected.
There are other hypothetical solutions used in science-fiction. Gravity control drives and field drives. The hand-waving explanation given for a gravity-control drive is that the fabric of space is bent, in an analogous manner to the way a gravitational field causes the curvature of space, and the entire vehicle, its contents, passengers and crew can be accelerated at any rate of acceleration including ultrahigh accelerations. Effectively the ship is in free fall in an artificially generated gravitational field. Exactly like people inside a lift (sorry, Americans, an elevator) falling down a lift shaft or spacecraft in orbit around a planet.
Field drives assume that every particle of a spacecraft is accelerated at the same uniform rate of acceleration relative to each other. This has the net effect of inside the spaceship of an apparent acceleration of zero. This is despite the fact the spaceship is accelerating across the solar system at, say, one hundred thousand g (100,000 g), but inside the vessel everything is weightless (because they're similarly in free fall). Although a thoughtful science-fiction writer will allow for a slight slippage in the field-drive's acceleration to allow an interior 1 g gravity for the comfort of passengers and crew.
By the way, that space capsule making a one-hour to Mars and had to travel twenty light minutes, had an acceleration of 100,000 g it would only have an acceleration phase of a bit over 100 seconds to attain a velocity of 100,000 km/sec. It will need to decelerate for a further 100 seconds at journey's end.
To travel to the outer solar system would either require accelerating close to lightspeed with ultrahigh acceleration drive-systems. For example, the planet Neptune is roughly four light hours distant, so getting there in hours means either accelerating and decelerating for the entire trip or choosing a high terminal velocity that is near lightspeed. For example, a trip at an average of half lightspeed will get you there in eight hours.
Possibly it may be necessary to resort to slow faster-than-light drives where a spacecraft travels at superluminal velocities of between 1.5 c and 4 c. The lower bound velocity gets to Neptune in two hours and forty minutes, and at the higher velocity in an hour flat. This does a pus-button type of FTL drive where you push a button and the FTL drive moves at a constant superluminal velocity. Stopping just requires pushing the off button.
Alternatively, a faster-than-light jump drive could be employed for quick trips across the solar system. This could have the design where the jump-drive requires "charging" its jump-engines for amount of time proportional to the distance it has to traverse.
Let's assume that the charging time is one-third of the distance in light time units. For example, the planet Saturn is one hour distant. A spaceship will charge its jump-engines for an entire twenty minutes. Then it can activate its jump-drive and instantaneously traverse the distance from Earth to Saturn (without passing through any points of spacetime between the two planets; aren't jump-drives wonderful like that).
This type of space travel requires extremely high accelerations or modest rates of FTL travel. Science-fiction is full of suitable examples.