One is to relax in the shower for a bit to soften your skin from the heat and moisture. Always shave your legs with shaving cream. Shaving dry legs will clog the razor with dry skin and will increase the chance of nicks.
When going to buy a razor, ask some friends on recommendations on razors.
Be sure to change razors or cartridges on a regular basis, a dull blade can lead to nicks. Also, it's best not to borrow your/another guy's razor. His hair is coarser and will dull a blade.
Exfoliate your skin before shaving. It gets rid of all the dead skin cells that could clog up your razor, preventing a close shave.
Since your leg hairs grow down, you'll want to start at your ankles and shave up. For your underarms, you'll need to shave in every direction since the hair there grows every which way.
When you done shaving, apply oil or moisturizer. The skin on your legs has few oil glands and has a tendency for dryness.
Ingrown hairs are caused when the hairs curl back under the skin. Avoid ingrowns by exfoliating daily and using a glycolic acid body lotion.