First off, I'm relatively new to the airbrush thing, so take all this with a grain of salt. BUT, I am the type to do tons and tons of research before making a purchase, so I feel like I can help.
As for the air pressure, I've read some say that 28-32 psi is the "sweet spot" for airbrushing minis, but I rarely push mine past about 22 or so. If you're doing terrain or vehicles, you can probably get use out of a higher psi. If you're working with minis, I find that anything about 22ish will be too strong and cause the paint to move around on the model. I use a TC-20T air compressor
http://www.tcpglobal.com/airbrushdepot/tc20t.html I think I paid maybe $150 ish for it? It's quiet and comes with a tank, moisture trap, and regulator -- all three of which you'll want. Here's the bummer -- it crapped out on me after about 3 weeks. BUT, the company really stands by their product. They sent me another one, and didn't even ask me to return the defective unit. I believe it comes with a 2 year warranty, too, so I feel pretty good about having it for a while.
Stay away from the canned air and pancake compressors. Both wastes of time and money. Some will suggest getting a heavy duty compressor at a hardware shop. They may be cheaper and have more power, but they are significantly louder. I have a really skittish kitty who gets scared of loud noises, and there's no way I could bring a loud air compressor into the house. That's why I went with the "hobby" compressor. A little bit pricier, but way quieter (the cat will even sleep on the recliner about 5 feet away while I airbrush. THAT'S quiet!)
As for airbrushes, there are tons and tons of airbrush guides out there. For minis, they all say the same thing -- internal mix, gravity feed, double action. Get one with all three. A lot of people swear by Iwata, but they're awfully pricey. I got a Paasche Talon that came with three different size needles and nozzles, so I can do relatively small detail work, as well as pretty large things like vehicles or terrain.
http://www.amazon.com/Paasche-TG-3F...QA9A/ref=sr_1_1?ie=UTF8&qid=1314364658&sr=8-1 There are a couple other good brands out there. I like that I can get replacement parts for my Paasche at a local art shop, which is pretty handy.
Also, be sure to get some sort of respirator to protect your lungs. The airbrush will atomize your paint and thinner, and you don't want to breath that stuff in (especially if you thin your paints with Windex, as many do, because the active ingredient that thins the paint is the ammonia). I got this bad boy
http://www.amazon.com/gp/product/B00009363G I won't win any style points for wearing it, but it keeps me safe. When I first started using the airbrush, I would skip wearing it for jobs I thought would "just take two seconds." They would inevitably take 10 minutes, and I would start getting headaches from breathing in all those fumes. Headaches stopped when I started wearing the mask religiously. Also, now I can pretend I'm Bane from The Dark Knight Rises.
Don't forget a cleaning station (I got this one
http://www.amazon.com/gp/product/B000VADIVC). It works for two really useful purposes: first, it's a convenient handle for your airbrush. This may seem silly, but you can't just lay your brush flat in the middle of a paint job or the paint will spill out of the cup. Second, it's a way to spray out your excess paint and run some cleaner through it before you move on to the next color. Just stick your gun in the hole (that's what she said) and spray it into the jar. There are also filters on the top of the jar that filter out all the paint fumes.
As far as cleaning the brush goes, you'll want to get some airbrush cleaner. But for big jobs, like cleaning the guck out of the whole gun (which you'll probably only do once in a while), I like the Ultrasonic Cleaner approach. Here's the one I got
http://www.amazon.com/gp/product/B000SOQ6KQ Fill it with Simple Green, turn on the ultrasonic thingy, and it will shake all the dried paint out of your gun. I've never tried this, but I bet it would make stripping paint off minis pretty easy, too. But I dunno.
Finally, I cannot recommend eye droppers enough!!
http://www.amazon.com/gp/product/B002J0NQCI If you try to drop your paint and thinner directly into your airbrush cup, you're going to run into trouble. First, it's harder to tell if you've thinned it enough (it needs to be thinner than you're used to -- you'll get a feel for it). Second, you'll inevitably be jamming a toothpick or stir stick into the airbrush to mix the thinner and paint. This could bend the needle, which is not good. I recommend mixing up your thinner and paint on your palette (I use one of those cheap-o, 99 cent well palettes from Michael's), then once you're sure it's thinned properly and free of foreign contaminants, you can suck it up with the eye droppers and transfer to your brush. I finally convinced myself to do this after I realized 90% of my clogs came from me taking the shortcut and trying to mix the paint directly in the airbrush. If you take the extra two seconds up front, you'll save yourself a ton of time taking apart your brush and cleaning out the clog.
Anyway, hope this is a good start. Many other painters much more experienced and talented than I am will probably come in and tell me how I'm doing everything wrong, but, like I said, I'm still learning myself!
The airbrush I was thinking one from the Iwata range. We'll see 