For just practicing you can go to any art supply store and get a bottle of acrylic airbrush paint. It's cheaper than model paint, works fine on paper, but doesn't stick to styrene worth a hoot.
As to whether to use acrylic or enamel on your models, that's your choice. There are benefits to both. Enamels generally adhere better and spray better but acrylics don't have the fumes associated with enamels and can be cleaned up with just water. The downside to acrylics is that, as the guy told you, they dry ridiculously fast. They can, in fact, dry in the air between the airbrush and the model surface. I personally spray acrylics almost exclusively. Not having a house full of paint fumes and being able to clean up with water is a big plus to me.
Thin your paint to about the consistency of milk, run your compressor down to about 15 psi, and spray from a few inches away from the surface. If the paint doesn't flow properly it may be too thick. If it's runny or won't cover it's too thin. If the surface is rough and you are using acrylics the paint is probably drying before it gets to the surface. A couple of drops of acrylic retarder, lowering the air pressure, or painting closer to the surface may help.
Regardless of what you use, I'd strongly recommend the use of a respirator. Paint fumes are just plain bad to breathe and a good respirator will be very beneficial in the long run.
Also keep in mind that the only way to learn to use an airbrush properly is lots and lots of practice. Nobody can do that for you, so don't get discouraged if it doesn't work well at first.