Empress and the rest of the people interested in airbrushing.
When you go looking for an airbrush, make sure it will do what you need it to do, also make sure it feels comfortable in your hands, if it doesn't, it doesn't matter how nice it is, or who makes it, you will end up either giving up or buying something else.
Make sure that whatever airbrush you buy, that you can get parts and service for it, and that it can take the use of the kind of paint or makeup you plan on using.
Keep in mind that siphon feed airbrushes require more airpressure to work than gravity feed airbrushes do, and that airbrush makeup uses from 5-12psi to spray, so when you go looking at airbrushes for doing makeup, that you realize that trying to balance the needs of the makeup with the needs of the airbrush to spray it. If you are buying an airbrush for makeup, you may wish to stick with a gravity feed airbrush, this lets gravity keep the paint flowing into the path of the air, and requires less airpressure to do it, this also means better control, and less overspray.
As for learning, use a water based acrylic paint to learn with. It is much cheaper than trying to learn with makeup, and a lot less tempormental. The basics for airbrushing are the same, regardless of the type of paint or makeup you will use in the end, and having to fight your paint while you are trying to master the airbrush is an extra layer of frustration you don't need.
As for the specs for the compressor, look at whatever the biggest tool or consumer of air and air pressure are, the airbrush it self doesn't require that much air pressure, at the most, about 65PSI per airbrush, and maybe 1.5CFM. These are for continious use, not peak ratings. If you want to use more than one airbrush at a time, then you need to factor that in. When in doubt, go with the biggest compressor you can get, with the greatest capacity, volume, automatic shutoff, lowest noise, one that is serviceable, and can fit in your garage, shed, etc..
There are a whole lot of companies out there that produce or sell makeup for use with the airbrush, the quality levels are all over the place, and the consistancy and prices can vary greatly. Do a lot of asking before you do any buying. It is so easy to get sucked into a sales pitch for stuff.