You can use just about anything that will give you a constant air supply at about 25-30 psi.
Although there are specific airbrush compressors, unless it is essential that they be quiet - like you're painting indoors, or in someone else's apartment - then it's not a concern, and any compressor will do.
As for practice; printer paper will do, but there are many other options.
A good one is to copy the practice sheets at airbrush tutor, get yourself an old picture frame with the glass in it, put the sheet you wish to practice in the frame, then paint on the glass.
You can get these from any charity shop for a couple of dollars.
Your sheet then becomes re-useable, and you learn to control your brush.
If you can do these exercises sucessfully on the glass, you can paint anything!
And at the end of it, just clean off the glass, and your good to go again.
At first, try an air pressure 25 -30 psi, and reduce paint about 5 to 1.
See what sort of lines you get, then maybe reduce some more.
There are heaps of tutorials and vids to get you started, but I recommend you do the one's from here, first.