Step 1: What you need:
You can use any kind of nontoxic metal wire, though I prefer to by jewelry grade wire online. Wire Sculpture and Rio Grande are excellent sources. If you're just beginning I recommend copper or bronze. They're inexpensive and look great with polish or patina. To get started buy at least 5-10 feet of 21 or 22 gauge dead soft SQUARE wire and 5 feet of 22 gauge dead soft HALF ROUND wire.
That's it, assuming you have something to wrap. Wrapping a round or oval cabochon in the 30 - 40 mm range will be easiest.
(Shown above: copper wire on a goldstone cabochon.)
Tools are very important. I used cheap jewelry pliers (~$10) for a while, but once I was sure I would stay with wire wrapping, I got good ones (~$50). It made a huge difference, especially in how tired my hands would get. If you do this, hang on to your cheap tools - there's always a use pliers you won't worry about. The basic tools you will need to get started are:
Flat nose pliers
Round nose pliers
Small round mandrel, bail pliers or pen
Later you can add chain nose pliers, jewelry files, bail making pliers and more. I treat myself to a new high-quality tool every so often.