Angela,
The first thing you will want to do is make sure your nails are primed and ready. Clean your nails, use nail polish remover to remove all traces of old polish. Applying polish on old polish leads to messy nails.
You can apply the way I was taught-base coat, allow to completly dry, use one later of polish, allow to completely dry, repeat colored coat, allow to dry, and finish with a top coat. I don't recommend it. It takes a long time.
Revlon used to make a polish that I had great luck with -- Revlon Colorstay. The colors were beautiful and it stayed on; Revlon discontinued this polish so you can only find it on eBay
Sally Hansen has their
Hard as Nails collection that has staying power. They also have a Hard as Nails topcoat that extends polish.
OPI does well for me if I use just one or two light coats--more than that it seems to chip.
Avon nail polish has been recommended as long lasting; I haven't tried this yet.
Essie is another great polish-just use a coat or a second light coat. Add a topcoat like Sally Hansen.
The key is to let it dry completely. I don't leave the house without nail polish and it drives me crazy when it chips too
My Blog: 
