What kind of airbrush? Double-action or single-action?
Could be your air pressure. At this scale, you'll probably want to shoot at no more than 15-20 PSI for minis, but you generally need to thin really well to spray at that pressure without clogging. The thicker the paint and the more impurities, the more likely you are to get clogs at low PSI. Some people will compensate by simply cranking up the pressure to force the crap through the nozzle, but generally not a good idea. Get your paint to the right consistency, don't force it through with more pressure.
Acrylic paints can be thinned with just water. I'm not sure what's in Vallejo thinner, but it's probably just water and maybe some acrylic mediums, possibly an extender. You want your paint to be about the consistency of whole milk. You don't want it water-thin, but you definitely don't want it too thick. You also don't want chunks of paint. You can strain the paint through cheese cloth or a fine metal strainer to get rid of congealed bits of paint, or you can buy an electric stirrer that will help make sure that everything is thoroughly mixed.
Something like this is great for mixing paint:
http://www.amazon.com/Badger-Air-Brush-Co-121-Paint/dp/B000BROV02/ref=cm_cr_pr_product_top
If you're using a single-action airbrush, it could be that you've got the nozzle closed down most of the way. Open it up and let a little more paint through and that might address the problem. It's also possible you used paint that was too thick and clogged up the works, in which case, you're probably going to need to take it apart and clean it (something you need to learn to do anyway). Taking apart an airbrush is actually pretty easy - if you search YouTube for your manufacturer and model number, there's a very good chance you'll find a video showing you how to take it apart and do a full cleaning. Ordinarily, you'll only need to do a full cleaning about once a month. Most of the time, just cleaning the cup, nozzle, and needle after use or between colors will be plenty.
But if you get crap stuck in the works, you have to do it. Metallic paints, pearlized paints, paints with glitter, anything along those lines are more likely to gum up the works than straight colors.
Also, you can use regular heavy body acrylic paints watered down. In the long run, that's going to be cheaper than mini paints, and at least as good quality if you buy good paints.
For practicing, go get yourself some craft paints and a pad of newsprint paper. Thin down the paints and practice, practice, practice. You need to get a feel for how much paint goes through the gun at different pressures and settings. Using an airbrush on minis is hard, but getting a feel for how much paint and how to aim will help. Newsprint is much cheaper than minis.
Not sure what else. There are lots of videos around YouTube showing how to use airbrushes. I'd highly recommend watching a few, and then it's just practice, practice, practice.
