Due to the large amount of energy required, the standard method for leaving the Earth's atmosphere is rocket propulsion. Is there any other methods in development that don't involve rockets? E.g. solar sails, space elevators?
Currently existing, and for leaving the atmosphere? No. It's always a rocket engine of one kind of another, whether for vertical launch or horizontal, solid fuel or oxygen-hydrogen.
Developed - definitely yes.
Solar sail is only viable in space, considering solar wind it catches is a miniscule fraction of physical (air) wind force, meaning it would be torn to shreds in the upper atmosphere and never provide enough thrust to overcome earth gravity. Same for the ion engine which has minimal thrust nowhere near to overcoming earth gravity, but can produce that thrust for years at a time using up minimal amounts of energy and fuel.
But there is a number of planned methods of launch that wouldn't involve rocket engines. There are already a few groups working on the space elevator plans - we're quite far from it, because the only viable material - nanotubes - despite dirt cheap raw materials, are still prohibitively expensive to manufacture. There was a plan to use antarctic ice to build a launch tube similar to multi-stage cannon, to launch vehicles diagonally using land-bound explosive charges; that was stalled due to environmental concerns. There was a plan for a "partial space elevator" - a rope orbiting the Earth at LEO, with trailing end reachable by vehicles with air-breathing jet engines, docking their payload to it, then the payload lifted to the orbit and launching from the other end of the rope, using space propulsion (solar sail, ion drive) from there.
The problem with these all is that while operating them would be significantly cheaper than rocket engines - per-launch cost a fraction of current, their initial cost outweighs the cost of a rocket-powered vehicle many times over. With current economy, they are all on hold or proceeding at glacial pace with only minor, inexpensive side tasks completed, while the bulk of the work awaits funding.
Multiple posters have mentioned but dismissed beanstalks because materials science isn't up to producing a sufficiently strong cable. So far nobody has mentioned the other ladder into space approach--launch loops. They don't require super materials because they are supported by objects moving faster than orbital velocity. The big downside is that they are supported by moving objects--if they quit moving they fall.
The basic idea is you build two stations that will pitch iron bars back and forth. You use extremely strong magnets to turn the bars around, once the bars are moving the energy input is minimal. They originally fly back and forth on a ballistic trajectory.
Then you build a track on top of the flying bars. The track deflects the bars downward, getting lift in the process. The speed of the bars is upped to compensate--they continue to follow the same course. Now you have what amounts to a train track that extends into space. There's no upper limit on the speed of a maglev train in vacuum.
Technically possible: Yes. Cost: Tremendous Safety: Not for me!
There's also the combination of the two ideas.
Build a set of towers in a ring around the Earth. They support a tube, inside the tube you have an iron ring. Spin the ring above orbital velocity, generate enough upward force to counteract the weight of the towers. Since we have countered the load of the towers we are free to build higher. Then build another ring to take the load. Repeat until you are as high as you want to go. If you're going past geosync the rings turn below orbital velocity in order to generate downward force rather than upward force.
Technically possible: Yes. You can build the rings and towers as close as needed to cope with material limits. Cost: Tremendous^2. Safety: While there are moving parts it's not like the launch loop that requires a turning station.
The key issue is, how do I accelerate this 'thing' to orbital velocity.
To date, only rockets have succeeded. Rockets are pretty efficient at generating high thrust with high impulse but are complex.
Other ideas touted in the part are thing like Orion, which used a giant vehicle, with a large plate, which on the downward facing side would detonate a nuclear device. Then the plate would absorb the impact and connected via springs (huge as your would imagine) would transfer that momentum to the vehicle. Clearly this is never going to happen. (Of course, the plot of the novell Footfall is that to repel an alien invasion, even Orion is acceptable). A rocket engine with no combustion, just riding the crest of nuclear explosions.
Other ideas are to use a high powered laser to heat a working fluid in the engine, so there is no rocket engine, combusting anything, rather the heated fluid exits the engine at high speed, transferring momentum. A rocket engine, with no combustion really.
Orbital beanstalks, a cable from a base in GeoSync, dropping down to the surface of the Earth is an idea, and it is still science fiction for the most part. (Cable needs to be 22,300 miles long. Enough to wrap several times around the Earth. It need a counterbalance weight or cable the same length sticking out the other way). But this would be an approach to reaching orbit without a rocket.
Railguns, or high energy chemical guns are similar in approach, accelerate the vehicle in a gun. Railguns work either by magnets (linear accelerator) or by vaporizing something with the current that generates the thrust. Chemical guns light the fire behind the vehicle, but it is pushed by the edge of the explosion (Sort of confined, chemical Orion).
Alas, no anti-gravity or neat trick to get to orbit.
To get out of the atmosphere in any useful way, you have to accelerate to at least the velocity associated with a low Earth orbit, and do so without falling back to the ground before that velocity is achieved.
Ion drives will almost certainly never do this. They are much more efficient with propellant (much less propellant mass expended to achieve a given delta-V) than a chemical rocket, but cannot be built to deliver the necessary vehicle acceleration. You need at least 1g to balance surface gravity, and then some to get you moving up and away. Ion engines deliver very small fractions of a g; they work well for spacecraft already in space on long-term missions, but are not considered to ever be a space launch technology.
There are a couple of interesting technologies which could be applied to space launch.
One is the SABRE engine. If successful, it could replace vertical pad-launch rocket with runway take-off almost like a conventional airliner for access to orbit. Although effectively a rocket engine at high altitudes where there is insuffucient air to operate a jet engine, SABRE would operate as an air-breathing jet at lower altitudes, substantially reducing the mass of propellant the vehicle would require at lift of for flight to orbit, as compared to conventional rocket.
Another is the nuclear rocket. This was pioneerd in the 50s and 60s, pursued to the point where a feasible launch system could be built, although non ever was. The nuclear rocket replaces chemical reactions with nuclear reactions as the power source to produce a stream of hot gas for propulsion. The exhaust velocity would be much higher than that of any chemical rocket, significantly reducing the mass fraction of the vehicle's propellant.
JP Aerospace, which currently has the altitude record for airships at 95,085 feet, plans to fly airships to orbit. Once boyancy has taken it above almost all of the atmosphere, it will however use a rocket engine of sorts, an electrochemical propulsion which during several hours accelerates it to orbital velocity. Wikipedia has an overview of orbital airships and here's a lengthy radio interview with John Powell of JP Aerospace.
The critical issue isn't height but Speed. Could a combination of precooled jets, ion drives and rockets enable us to journey into space cheaper, safer and more efficiently than the current rocket based method? Using a precooled jet engine to gain altitude then using an ion drive to give extra thrust you could achieve the required velocity. Reversing the process you could land the craft conventionally removing the need for heat shields and other hefty re-entry devices. http://youtu.be/KFL623O9CXQ