Well, there are plenty of in-depth guides by smarter people than me who can give you more detail than I can, but I'll take a stab at an overview:
First off, you need primer. Most people use a spray can with oil-based paint, but there are plenty of alternatives if you don't like that idea. Most people will also tell you NOT to buy a high-end brand-name primer. Testors (compared to GW's spray primer) is cheap and effective, from what I'm told (I don't spray-prime). Whatever brand you choose, the main colors are black, white and gray. If you are planning a light color scheme (lots of reds and yellows), then prime white. For daker schemes (black, purple, deep blue, etc.) prime black. Not sure, or planning to do both? Get gray.
As to which paints, the best advice is to make sure you get model paints. Sounds like a given, but if you don't know, there IS a difference. Model paint is designed to have a high pigment in thin coats.
Speaking of thinning, get yourself an eyedropper or an empty bottle with a dropper tip (Vallejo makes them), and get used to using it to add water (or thinner, if working with oil paints) to thin them down. It means you'll often need to do more than one coat, but in the end it will be worth it.
Most model paints are acrylic, so they thin with water, and clean up with water, too (although you'll never get them out of fabric or carpet -- beware!).
As to brands, there are tons, but here's what little I can tell you, based on experience and knowledge from others, about the Big Brands(tm):
GW/Citadel Paints: If you're playing games workshop games (40k, WHFB, LoTR), you've likely heard of these. You're probably also likely to find them wherever you find the models. They're not bad, and a good start. Personally, I find other brands cover a little better with "regular" colors, but of note is Citadel's Foundation range, which covers nicely over primer (whatever color the primer and paint is) and they work well as basecoat colors. Also good are the wash sets, which make shading MUCH easier. I also like Citadel's metallics, and have heard (but can't confirm) that these metallics work better than other brands.
Vallejo: The second most popular I've seen. They come in dropper-tip bottles, which makes mixing colors a breeze. They also tend to run thinner on average than Citadel, though you should still get used to thinning your paints. They have two ranges that combine to offer MANY more colors than Citadel, and one range (their Game Color series) was clearly made to compete directly with Citadel (including "rip-off" color names). Vallejo also makes mediums like glaze, matt, etc., that Citadel doesn't make at all. The "regular" colors cover much better, I think, than Citadel. I know nothing, however, about Vallejo inks/washes or their metallics, but I hear they aren't as good as Citadel's.
Tamiya: I confess I don't use these, and can only offer anecdotal testimony. I hear of them most often used in air guns on vehicles. I've also heard that they don't work well with acrylic-based washes, so if you use them to basecoat, you should expect to use oil-based paints or products for the rest of the project (though this is WAY beyond my understanding).
Reaper: An old brand that has been revamped in recent years. I haven't tried them in a LONG time, but I hear they're pretty darned good. Like Vallejo, they tend to run thinner, and also come in dropper-tip bottles. Reaper also sells triple-bottle sets, with a base color, one lighter version, and one darker version, which makes the shading and highlighting stages much easier.
P3: A fairly new brand from Privateer Press. Like Reaper, they have highlight and shade versions of their colors, but they don't use dropper-tip bottles. Haven't used any of theirs, so I can't say much about them, but I've heard many good things.
Prices vary between $2-4(US)/bottle. I'm not sure which is cheapest, but I will say one bottle goes a LONG way, so they're worth the money. Also keep in mind that the above brands are made for miniature painting. Other model paints, such as Testors, are worth a look, though I can't comment on them. I also should point out that if you like oil paints, that's a whole different world, and I'll defer to others for opinions on that.
Good luck, and be sure to search the forums for more specific/better advice!