Make sure your nails are clean to start - if they have an oily soap residue or anything, the polish won't adhere well. A base coat can help with this.
I like OPI and Essie, and also have had good luck with the Sally Hanson Salon Manicures line.
I use a top coat from Essie - Good to Go. It really does help put a hard layer on top, my polish lasts a couple of weeks with this. Then I use a quick-dry from Essie, it actually comes with a little eye dropper and you place a drop or two on each nail. They're hard within seconds, best quick-dry I've found.
For painting:
If you want to get fancy, start by soaking your hands in warm water to soften the cuticle, then push them back with a wooden nail stick. You can skip this step if you don't want to bother with it, it just gives a "cleaner" finish.
To start with, sit at a desk or table, and place one hand on the table (once you get used to it, you can sit on a couch and place one hand on your knee).
I try to avoid putting my "painting" hand directly over a nail that I've just painted. I start by using my right hand to paint my left hand nails, starting with the pinky and working toward the thumb - that way my right hand doesn't accidentally bump anything that I've just painted. Then I use my left hand to paint my right hand, again starting with my pinky and working toward the thumb.
For each finger: dip the brush into the polish, and slowly drag it up one side to clean some of the polish off of the brush. There should still be quite a bit on, though. Start by running the brush down the center of the nail, then take another swipe down each side. If it's a good polish, the swipes should melt together (polish can get gummy when it's too old, and will look streaky no matter what you do).
Let each coat dry for a minute or so between layers. Usually, by the time I finish the second hand, the first pinky is ready to go again.
Start with a neutral color, graduate to dark colors after you get a feel for it. And have fun!