If you get a quote from an insurance broker, you can call on your cell phone with your credit card and get insured on the spot after you buy it. If your family already has insurance, it may also be temporarily covered under the grace period for that policy.
The seller will have a title, which probably has paperwork on the back that he/she fills out to transfer the car to you. You file your half with any applicable fees, he files his, and the car is yours. The DMV will issue a new title in your name. Some states may also have rules about who needs to smog-check the car. California requires the seller to smog the car no more than 90 days before the sale.
I also second alkemyst's suggestion to do the transaction at your DMV so both you and the buyer know the paperwork goes through properly. It protects both of you.
Bring cash. No smart seller accepts anything else in these transactions, unless it is a really expensive car and you both go to the bank together to get a cashier's check in person.
If your target car is a popular model, have the cash ready before you start calling people, because some cars will be sold within 24 hours and you will not have time to get your funds together. When I bought my fiancee's Accord, the ad was only 16 hours old and I beat another buyer by only 30 minutes or so (he was on his way and trying to bid up the price over the phone, but luckily the seller preferred to just get it over with).