I remember looking into this when I played way too much Kerbal Space Program and got sufficiently involved that I installed the fully realistic version of the game.
The accepted answer regarding aerobreaking is pretty good and covers the basics, but there's some more stuff going on that I can elaborate on and I'd like to. That answer's assertion that the technology doesn't exist yet is true, but there's also some really exciting technology on the horizon and this answer will rundown some of the methods that you'd need to do some of the things you want.
Context - What is High Altitude Flight?
Broadly, entering and leaving an atmosphere are synonymous with flying through it at any height and altitude you want. So, let's define a few terms.
The first thing is the speed of sound in air, called Mach. Usually around 330m/s but it varies with temperature and pressure. Mach 1 refers to flying at the speed of sound.
When flying at the speed of sound, the way air behaves changes and this results in the air compressing for the first time. When it compresses it heats up. Were the heat to touch the surface of a craft, it could melt the craft. Spacecraft during re-entry are specifically built (not entirely true, but close enough) NOT to fly. They don't cut through the air like a plane, they do it with a very unaerodynamic surface that creates a bow-shock that keeps the heat away from the craft. Therefore, at the moment, all re-entries don't entirely count as controlled (again, not entirely true, but good enough for now). Even the space shuttle doesn't enter like an aircraft, it enters at a 40 degree angle to the flight direction until it slows down safely enough to fly. The fastest flying plane (the SR-71 Blackbird) only got up to Mach 2 and that had to be built so that it leaked fuel on the ground to allow heat expansion caused by flying at high speed.
Orbital speed is about Mach 25ish, a long way past Mach 2. Attain that in the atmosphere and you stop flying and start lifting because you're actually orbiting. The good news from Kerbal Space Program is that, if you could fly into orbit within the atmosphere (Kerbin, the Earth-like planet in KSP only has an escape velocity of Mach 8, so this is easily doable), you can get into orbit with ridiculously low amounts of fuel. It's probably the most fuel efficient way to get there and it would probably also work on the Earth, but there are lots of problems to solve, which I'll come back to later.
First, however, I also have to talk about the atmosphere, because that's also important. For one thing, the atmosphere is thickest near the surface but the density of the atmosphere drops off exponentially the further you go away from the earth. This is both good and bad.
It's bad, because less atmosphere means less lift and less air for the engines to use. This sets a high ceiling on what's safe for aircraft to fly. The height for jet engines to keep in mind is about 26km. After that, there's not enough air to keep flying.
The good thing, however, is that there's also less air to cause drag and cause heating effects. Get high enough up into the atmosphere and you can go faster. Go faster and you can scoop up more air for use in your engines, keeping the effective density high enough to fly. We can't do this yet, but it can be done.
Because air density falls off exponentially, it's hard to say where the atmosphere finishes. The International Space Station is technically orbiting inside the Earth's atmosphere, but isn't really for most practical purposes. The number to think of for most purposes seems to be 100km.
Reframing the Problem
So, your question is nearly identical to - how to I get a plane to fly at speeds between Mach 2-25 at altitudes of 26km or above?
Well, the answer is that we can't, but we're working on it.
Jet Engines and Beyond
Jet engines work on a single principle that air speeds up as it is compressed below Mach 1 and then speeds up as it expands, leaving the jet engine at greater than Mach 1. This different behaviour below and above the Mach line is what causes the jet engine to work. Jet engines do this by sucking air in, compressing it and then heating it, spewing it out the back.
In theory, you'd think that the jet engine ceases to work functionally at speeds greater than Mach 1, but you'd be wrong. Air slows down as it enters the engine, compressing and heating itself. The effect becomes so pronounced that, were you to get up to high speeds, the turbine part of the jet engine becomes unnecessary. Air is getting sucked in and heated up anyway, you just need to add fuel and let it burn.
This type of engine is called a Ramjet and, well, we've kinda built them, because the SR-71's engines actually worked that way past a certain speed. Basically, you stick the afterburners on and the engine works, with the turbines acting as drag.
This works so well that there's a real problem. The engine overheats from the air coming in above a certain velocity and the engine explodes. This is bad. However, a British team, in trying to develop a hybrid rocket/jet engine called the SABRE have solved this and can super-cool the air and recycle the heat into the fuel, allowing jet engines to push up to a theoretical Mach 4/5. (The super-cooler exists, the engine doesn't yet, but funding has been approved, search Reaction Engines, because it's awesome.)
Above Mach 4/5, something happens to the air coming in. The air stops slowing down enough to flow below the speed of sound and instead the whole airflow is faster than the speed of sound. We go from a Ramjet to a Scramjet.
Scramjets work on the fact that the air coming in is being compressed and heating up, like in the Ramjet, so adding fuel causes it to expand more and give more thrust. It's not as efficient as a Ramjet, but good enough and is better then a rocket engine until you hit Mach 10, in theory.
Research in Scramjets is ongoing. Of the Scramjets known to exist, NASA/US Military have built one. Because of the nature of the research, it's military sensitive and therefore no one knows what happened or if it was successful. Also, the internet knows that the Chinese do not have one and did not complete tests recently (so don't ask, OK?). These seems to show speeds of Mach 6 and that the tech is possible, though why it's not being used to make cheap ICBMs is anyone's guess.
Finally, it's worth mentioning rocket engines, which carry their own reaction mass, so don't need to grab it from the atmosphere. These work (approximately) as well where-ever they are used (though the more atmosphere you have, the worse the thrust out the back is).
Flying High
So, the biggest problem with flying fast is heat and also dynamic pressure. Dynamic pressure is what the air exerts on the aircraft as it goes through it. The number to keep in your head, which is why the Space Shuttle launch system actually gets throttled back during launch, is 70,000 KPascal. More then this, you get lovely things like aerodynamical failure (aka the polite KSP term for the wings ripped off). This is bad.
You can derive that using really simple physics using a momentum transfer argument (which I invite you to either do or search for) and you'll notice it related to density, so flying fast in the upper atmosphere is not so much of a problem (though you do get heating because it's faster than the speed of sound up there).
In the lower atmosphere, it's incredibly bad and flying at close to Mach 1 close to the ground in KSP got me to 40,000kPascals near the ground (and wings ripping off, did I mention the wings ripping off?). Higher up, Mach 4/5 was not a problem.
The faster you fly, the more heat you generate and the faster you need to fly to maintain air-pressure in your engines. At the moment, heat dissipation inside engines far outstrips the ability of engines to cope with them. Since that's being solved, after that, it seems like heat to the airframe is the next problem (Mach 4-6) and both of these limit the speed at which you can go and also the height (since you can't fly fast enough to get enough air into the engines to go higher).
For this reason, the Skylon Concept, based off the Reaction Engines SABRE hybrid engine, flies up to 26km at Mach 4/5 and makes like a rocket after that.
Beyond Mach 6. I don't think anyone really knows what happens to airflow. Except NASA. They get up to those speeds on launches. But there's some odd things happening such as when supersonic air boundaries collide (this causes the rocket exhaust, which is like a jet of super-fast air, coming out behind the rocket, to widen, because it's interacting with the shock cone of air from the rocket's nose). In a feasibility study of the Skylon, the only problem NASA found with the design was that, with the engines in the middle of the craft, the rocket plumes could expand and cook the tail of the aircraft at greater then Mach 12, though a small bit of rejigging of the engine position could get the Skylon past that, but with worse control.
So, in terms of aircraft that are in the near-future pipeline, the Skylon's design is controlled flight below 26km, but at Mach 4/5, and rocketed flight beyond, which is an improvement on anything that exists. However, also the proposed SR-72 is supposed to be Scramjet driven and no one knows what it's specs will be (or whether it will be built), but if it does work and fly (remember Scramjets need a minimum of about Mach 4 to work), it will be the fastest and highest plane ever to exist.
Also worth mentioning is the Virgin Galactic plane, which uses a two stage system, with cheap, commercial flight up to a ceiling and then a rocket component from there on. Thanks to the system being two stage, the weight considerations of having two engine systems don't matter (weight considerations mean that carrying two engines is extremely uneconomical in most cases).
Re-entry
OK, this is maybe a personal opinion from playing too much KSP, but we're doing re-entry all wrong. So, I wrote that it's perfectly fine to fly up in the higher atmosphere and, well, it seems to work for me - if I have fuel.
I mentioned that all current re-entries use a bow-shock system. This causes the heat compression wave of super-high Mach speeds to stay away from touching the aircraft. This stops the aircraft from melting and, worse, exploding.
The reason this is done is that it's really, really hard to get fuel into space, so most space-craft are landing without any fuel on board. This means they're all effectively gliders.
If you have the technology for advanced spaceflight (which we'll assume for this section), you really don't want to do that. In fact, it's super-easy to make a flying transition in or even use lots of air-brakes (which again space-craft don't carry because of weight). I designed lots of space-planes in KSP that did just that and even managed to do it with a Skylon prototype in super-realism mode when it didn't quite get up to orbital speed, but close (I never did get into orbital on super-realism mode with a space-plane, I was always a Mach number or two short, make of that what you will).
I'm going to guess that the realistic way for now is purely done that way because of weight considerations and that the bow-shock method is the safest and best gliding method to use when every kilo counts (which is does with rockets). (I never could safely pull off a gliding re-entry even on the forgiving Kerbin Earth-analogue, that wasn't bow-shocked and these have to be steep to avoid bouncing off the atmosphere in the initial stages.)
If you have any thrust left on re-entry (preferably in rocket fuel, that doesn't need the atmosphere to generate thrust) and some wing-span, you can easily skip over the outer surface of the atmosphere, bleeding speed off gently until you fall into safe flight regimes for entering the lower atmosphere (I did this a lot when I missed the space center on re-entry). However, you might want to take this section with a bit more skepticism than the others.
The Shape of Things - Aircraft Design
Something that's also known, if you get into aircraft design, is that air-craft designs vary based on what speed the aircraft is designed to fly at (the flight regime).
If you see a glider, that's the optimal shape for low-speed flight. Big, wide straight wings and a body that's even vaguely aerodynamic will do.
Commercial aircraft are good designs for aircraft flying at speeds less than Mach 1 but close to it. These involve big, slightly canted and slightly swept back wings.
Super-sonic aircraft have to have very swept profiles to dissipate heating and pressure during super-sonic flight and are very sleek.
Hyper-sonic aircraft like to look like rockets, more or less, with very little in the way of wings and these so swept back as to avoid heat and pressure issues, with the body also extremely sleek and ballistic.
The problem with designing spaceplanes is that you have to design a plane that can take off and land (so flying really slowly and at low altitude), fly a super-sonic speeds and make like a rocket.
The Concorde was an example of how to fly at low and super-sonic speeds and the design was so bad at low altitude flight that, was it not for a trick involving delta wings, the thing could never have landed at a commercial airport (this accounts for its super-sized forward landing gear).
As it is, most military super-sonic jets require longer runways to take off and land on, so that the plane can get up to speed before taking off, and will use parachutes for braking.
The Skylon design, for example, is so bad at flying it needs a 3km runway and will have a water cooling system for braking if it has to abort, that's how fast it'll be going on the ground and how bad it is at flying at low speeds.
In this one way, flying into the atmosphere and flying back out is actually easier than designing aircraft for use in all the atmosphere, since you don't have to worry about pesky lift issues for landing/take-off and you can keep a profile of very small and swept back wings ideal for flight at high speeds. Or, you could use a self-lifting body, like NASA did for the Scramject test.
Conclusion
We can't do a lot of this stuff, but we're working on it. It's exciting and so check life for progress...
