I have some, but not alot, of experience with the wonder of airbrushing. I use a double action brush and a simple "put-put" compressor (no tank or regulator). I've talked to a friend of mine recently who uses an airbrush alot. I gleamed this from him.
The compressor is only dependent on what you want to do. If you are just looking for a fast way to base coat, a compressor with out a tank is fine. Not having a tank means that when you press the button on the airbrush the pressure changes for a second and you can get inconsistent sprays. Having a tank, with a regulator, fixes that (something i will be investing in soon).
For the brush, everything depends on what you are using the brush for. If you are doing simple highlights or just adding a bit of blending to a big area like on a vehicle you can most likely get away with a single action. At the time he was telling me all this he was using a single action to add a black trim to the edges of some eldar tanks. However, if you are going with fine detail on models like a marine or using it to do a mural or some other really detailed piece, you really need a double action.
Since I am using it only for base coating and simple highlights on a large piece like vehicles, im fine with my setup, but i will need a tank compressor to do finer details.
Also, a place you may look for cheap compressors is Harbor Freight. Not sure on the airbrush, I got mine on sale plus a coupon at Michaels.
Again, I have a limited experience with air bushes, but I'm happy to share. Its a hard tool to use and takes a bit to learn.