I recommend starting out with a fairly inexpensive airbrush and after you get the hang of it if it's something you really like you can invest in a better model. A double action is definitely the way to go as it gives you much more control than a single action. This link has some great information on what to look for and what to buy:
http://www.goldenpaints.com/technicaldata/airtips.phpThe problem with compressed air is the air pressure is inconsistent and there's no way to regulate it. Also, if you plan on airbrushing more than a few times the cost of canned air can quickly add up to the cost of a compressor.
Any air compressor can be used with an airbrush although you'll want make sure you have a regulator and moisture trap (these features are built in to a compressor made for airbrushes). The regulator will allow you to adjust you psi (different psi for different types/thickness of paint) and will also trap any excess moisture in you line to prevent spider webbing. Most will look something like this...
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loginBefore I got my small airbrush compressor I used the one my husband has in the garage for his air tools. It was REALLY loud so one day he brought me home an airbrush compressor. I see absolutely no difference in the quality of my work but the compressor itself is much, much quieter so that's one advantage.