I started painting my nails about a year ago. That may not seem like very long, but it gives me an advantage because I still remember which things I had to learn and which were obvious. That means you won't have to see me say "now just do an inverted Woollongong Shimmy and you're finished!"
Beyond the obvious colored polish, there are a few supplies you should grab.
(All the Amazon links here are mostly intended as examples; they aren't necessarily the cheapest or best option)
Base coat: Base coat goes on first and serves two purposes: it helps the polish last longer without chipping, and it keeps the polish from staining your nails.
Top coat: as you might guess, top coat goes on last. Along with the base coat it helps prevent chipping, and it gives you a nice shiny finish. A matte top coat is another option that looks nice with solid colors.
Nail polish remover: If you decide you want a new color, or when the polish starts chipping, you'll want to take it off your nails. Polish remover is a usually-acetone-based solvent that...removes nail polish.
Q-tips (cotton swabs): It's common to get a bit of paint on the skin around your nails. Clean it off with a q-tip dipped in polish remover.
Aluminum foil: Acetone is a pretty vigorous solvent that can ruin the finish on your furniture. I do my work on a bit of foil so that doesn't happen.
Some additional tools will prove helpful if you want to do anything fancier than solid colors:
Striping brush: This is a brush with very long fibers, for making straight lines.
Fine brush: A small brush for making delicate designs.
Dotting tool: A metal rod with spheres on each end that makes dots. It's identical to the stylus you may have for carbon tracing or making lines in clay.
Cosmetic Sponges: Pieces of fine-pore synthetic foam are good for applying gradients (see below for more).
Wooden Pencil: A pencil eraser is good for making medium circles.
Finally, some things for your nails themselves:
Cutting tool: normal nail clippers will work ok. I find nail scissors are a little more precise.
File: Emery boards work, but they wear out fast. I highly recommend a crystal file, which is made of glass. They don't wear out at all, so you'll quickly recoup the upfront cost, and the cutting surfaces are sharper and finer than an emery board.
Nail Block: This is basically a file glued to a foam block. It's nice for shining up the surfaces of your nails, rather than the ends. It won't matter if you're painting, but if you're going out "bare" a nail block can give you a nice shine.
Whether you're doing fancy multi-colored art or a simple single color, the fundamentals are the same.
First of all, for the next hour or so, you aren't going to be able to use your hands. I usually turn on the TV so I have something to occupy my attention while I wait for each layer to dry. Go to the bathroom now!
Gather all your supplies: base and top coat, color, foil, and any tools. The basic nail paint is 4 coats: base, two layers of color, and topcoat. Let each coat dry for 10-15 minutes before you start the next one. While you wait for a coat to dry, you kinda have to pretend you're a surgeon and avoid touching anything. You can do a little bit with your fingertips, but it's surprisingly difficult to do anything productive. I'm overemphasizing this point because you still don't believe me, but you'll learn for yourself in time.
I've seen various tricks for preventing chips, like soaking your fingers in ice water after each coat. Their benefit is minimal; a good base- and top-coat are much more effective.
The best way to get a smooth coat of lacquer is to load a moderate amount onto your brush. Let a small drop fall onto the base of your nail, then spread it toward the tip. If it takes more than one pass to get a good layer, you don't have enough on the brush; if you have any blobs or pools, you have too much.
Don't worry about getting thick, fully-opaque color in one go--that's why you're doing two coats.
Too much paint on the brush.
Not enough paint on the brush.
Just the right amount.
Expensive vs. Cheap Lacquers
A bottle of nail paint can range from USD $0.50 at a grocery store endcap to $10.00 for high-quality brands like OPI. I haven't noticed a significant correlation between price and the quality or longevity of the color. However, good lacquers spread more smoothly and are much easier to use for delicate work.
If you aren't a skilled painter you might feel intimidated by the thought of trying to work in a tiny space, especially with your non-dominant hand. Don't be! I myself have such shaky hands that I've been accused of faking it. Nevertheless, I've had a lot of success with this one weird trick: go really slow. With a fine brush, a quality lacquer, and patience, even a text-oriented keyboard-masher like me can manage some nice designs.
Other Design Ideas
A nice way to go above and beyond a simple one-color job is to mix two colors into a gradient (or "ombre," if you prefer). You'll need makeup sponges and a couple of compatible colors. I've found that colors from different brands may or may not mix well; I assume this is a matter of differing solvent bases.
A marbled look takes some work, but looks really impressive when you pull it off. It's especially a good way to take things up a notch if you're intimidated by fine brushwork.
Simple dots or stripes take minimal effort and stand out in a nice way. Use a light touch with your dotting tool; pressing hard against the nails will make a sort of empty ring rather than a dot.
If you're growing your nails out as well as painting them, you'll discover a tendency to crack, chip, or peel. The only real way to prevent damage to your nails is to avoid using your hands, but that isn't very practical. Typing, in particular, seems hard on your nails.
There are lots of polish removers and base coats that allege to strengthen your nails. They don't work, so don't waste your money. I've been trying a biotin supplement, with some success. Switching to a non-acetone nail polish remover also seemed to help somewhat.