I brought cash to give the owner. He then signs over the title to you (if applicable -- In Maine, a vehicle over 15 years old doesn't need a title to sell, but your laws may vary) and gives you a bill of sale. Normally, the owner would remove his plates and you get temps from the police station or where ever they do it near you. Drive it home. In Maine, you buy insurance and bring it to the DMV or town office and pay taxes, title fee, registration, etc. You turn over the title to them and receive one with your name on it in the mail in a few weeks. Bring your insurance card -- you may need it to prove you have insurance. Each state varies with their laws. Mainers need insurance to register, and a vehicle newer than 2001 has MSRP value taxes applied to it. Depending on where you live and the fees involved, expect to pay about $200 for registration and fees. Again, it may be different where you live, but for me, that's how much it cost to register. You get your plates upon registering and you're set to go.
Hope this helps.