the compressor you currently have will be perfect for airbrushing. it has a mid sized storage tank for compressed air (so the motor isn't constantly running like it would be if you bought an airbrush specific compressor).
like Ben said, you pour the paint (which just comes in a plastic flip top bottle) into either one those two glass jars (the 1oz jar is the one attached to the airbrush in the picture on the box, the 3oz one is the bigger non-cylindrical one) or the metal paint cup. all three of this fit into the bottom of that paasche airbrush. you don't need to keep buying these jars (unless you want to), you can just clean them out and use new colors.
the paasche airbrush you linked to is single action meaning when you operate it, you can only push the trigger (on top of the brush) down, and this releases a steady stream of air at whatever psi you have your compressor set to. to adjust paint flow, you turn that little cone that is located under the tip of the brush (kind of at a 45° angle) to allow more or less paint to flow through and mix with your air. when the air shoots through the brush, it passes across the tube leading into your paint cup/jar, and by means of the Venturi effect (http://en.wikipedia.org/wiki/Venturi_effect), paint is sucked up the tube and mixes with the air, forming the "atomized" spray which evenly coats your lure.
that said...
it would be of your best interest to look for an inexpensive dual action gravity fed airbrush to start with. a dual action brush has a trigger that allows you to press down to allow air flow, and then you can gradually pull the trigger back to allow more paint to flow through. if you just push the trigger down, only air comes out of the brush, if you pull the trigger all the way back, a lot of paint sprays out. this dual action is better for two reasons 1.) it allows you to control paint flow on the fly more easily and 2.) it allows you greater control of detail with the brush.
gravity fed means that the paint cup is fixed (though sometimes they are removable) on top of the brush, like the one Ben linked you to. this has a few advantages as well. 1.) you can shoot very small amounts of paint, like one or two drops at a time, and minimize wasted paint, 2.) it is easier to clean because there isn't a little tube that transfers the paint to the airstream (on a gravity fed brush you need to clean the inside of this tube or your colors will mix or it gets clogged).
like Ben said, you should probably start with Createx or any other water based airbrush-specific paint (Smith Wildlife Colors, Poly Transpar, Auto Airs are a few others). its just simple water cleanup and there are no hazardous solvents and fumes to worry about.
paint never gets in the hose, that only transfers air from your compressor to your brush. to clean out the paint cup and the tube that transfers paint to the brush, use water in a little squirt bottle and a small gentle brush. they sell airbrush cleaning brushes.
thats about all i can think of at the moment. hope i didn't bore you or tl;dr. airbrushes are simple machines once you understand them.
