I prefer a tank-style compressor; however, the Central Machinery compressor from harbor freight I have worked fine for my airbrush, but you can't pump up a tire or rum air tools with it.
If you do not need to pump up tires, drive nails, or run air-powered tools get the Harbor Freight compressor.
Again, I highly recommend the Badger Anthem 155 airbrushes. These airbrushes will spray anything you are likely to want to spray. They are repairable, and parts are readily available from badger and other sources. Badger airbrushes are made in the USA. Lowcost China airbrushes can work for you, but I treat them as disposable. If they break, getting parts is a crap shoot, and you may not be able to find them.
You are also going to need some accessories. Paint jars and and quick disconnects come to mind.