Colors are all up to you. Look around to what people have done. Use something simple like MicroSoft Paint or even Excel / Word to doodle. Also up to your budget and intentions.
Powder coating is the way to go, but costs a pretty penny, too. Using paint (not powder) on certain areas simply won't last if you plan on trail riding. Engine parts get hot and oily, and regular paint will peel quickly. A-arms get pummeled by brush and stones. So if you paint it and ride it, expect to touch it up. Find hi-temp engine or exhaust paint, but remember that some of them have a bake cycle (read the directions).
Powder just last longer, as you are baking / melting a powder onto the surface and all the pores, crevices, and surface blemishes. And it cures harder and heavier than paint. Powder coating isn't hard to do yourself on a small scale. You can get a system for under $100, but then need an old kitchen stove (oven) for baking.
But I imagine most people will paint since it's easy, it looks good, and it's affordable. Just be sure to prep surfaces really good. Just be sure to prep surfaces really good. And just be sure to prep surfaces really good. (did I make myself clear? ha) If you can sand or bead blast all the old coatings away, that's your best bet. Then clean it up good with some degreaser or soap/water (or some kind of an solution acid dip if you know anyone with a setup). Wipe it clean with mineral spirits, isopropyl alchohol, or brake cleaner. Paint with patience. Multiple thin coats, not 1-2 thick coats. Let it dry. Don't touch it. In fact, bake it if you can with an oven or even a heat gun.
There are some textured powder coatings, too. Orange peel, crackle, etc. And I've seen some a-arms and frames even painted with crap like rhino-lining (bedliner).
For quad racers, I always stick with blue frame, gray / silver a-arms, blue bars, black steer stem and tie rods, natural swing arm, silver rad screen, and then the engine blue with semi-polished / natural head and cases. It flows nice, looks like stock, and looks great with the yellow fenders and decals. I've seen some yellow frames and yellow engines and it...looks awful IMO. And the all-blue engine is original, but boring. It doesn't take much to make the case covers and head look cool (bead blast, wire wheel, light polish).