1: Aircraft Remover! Aka airplane grade chemical paint stripper. Comes as a gel, wipe it on with a rag, leave it on til the paint is completely bubbled, about 20-30 minutes, then wipe it off with a few different rags and all the paint will be completely gone. But WEAR GLOVES! This crap burns like hell for a while when it touches your skin. And try not to breathe it in either.
2: Clean the frame or parts VERY thoroughly with paint thinner or alcohol. Cuz if ANY dirt or oil from your fingerprints or remover is left on there, it will mess up the paint before it even goes on.
3: Next, use masking tape to tape off the inside of your headtube and bottom bracket. That way no paint gets in, possibly messing up the way that bearing rest in them, plus it makes it easier to get the paint right up to the edge of the bearing, making it look more like the frame came that way.
4: Pretty much any kind of paint. I mostly use the cheap Wal Mart 99cent cans. They don't have a very wide selection for colors, but you can get really good results if you know what you're doing.
5: Be sure to actually give yourself space to work. Aka, hang your parts/frame from the roof of your garage or where ever for best results, that way you have 360 degrees of room to work and won't have to touch anything, possibly messing it up. This is what I do and it helps ALOT.
6: Lay down a couple base coats. Try to stay between 8 and 12 inches from whatever you're painting. That way it covers it, but doesn't get such a heavy coat over it that it starts to drip. Cuz once a drip starts, you can't stop it and it will completely ruin a paintjob. Once you've left it to dry for a long time(hours, not minutes), sand the entire thing lightly with a very fine sandpaper. This way when you start laying down actual coats, the paint won't flake off the first time you or something hits it, and will make your paintjob last ALOT longer than just quick spraying it. And spray paint is particles of air and paint, so try to do this in your garage or somewhere where there's absolutly no wind blowing for obvious reasons. Now wipe everything off with a CLEAN rag so there's no sanding dust, cuz dust will once again screw everything up.
7: Now start laying down actual color coats. Once again staying a reasonable distance to avoid drips. Spray 2 or 3 light coats(or more if you'd like), about an hour or two apart, then apply 2 coats(or more if you'd like) of a spray clear coat, once again waiting in between coats. Then when you're done, leave the parts or frame to hang for about 48 hours. I know it's hard to wait, but if you don't, the first time you hit the frame, the paint will get destroyed and all of this will have been a waste of time.
8: Once drying time is over, take your parts or frame down and throw everything back together, but be carful, cuz your paintjob is still fragile at this point. Just be careful not to hit it within the next couple days and you'll be good.
Refs: Marco_Gonzalez, Scramhype, Russellonabike, Dee Hos, Lbonbi, Pcoco, Raz, Bug31, Wallywangois2hip, Paul Hutchison