In reply to Tarquell:
Echoing what StevieB and SNC said - you can do some of it yourself, but it's an out-and-back walk - you can see down into it in several places, though.
The path is ok, I did it in sandals at exactly this time of year. Bear in mind that it can get quite chilly/windy at the top, even if it's hot at the bottom, so take a jumper just in case.
The tourist kiosks will obviously tell you it's far too dangerous to drive up yourself, but it's fine - they take coaches up there! Just be careful there isn't one coming the other way as you go round the hairpins. As SNC said, the car park is organised chaos - be prepared to wait for a parking space while the chap directs traffic in typically Italian fashion.
There's a souvenir/drinks kiosk in the car park and another along the rim - with prices to match the altitude, iirc.
There's another car park where you can walk up from - see the details in the link Mark posted.
If you have time for a day trip to Capri, you can get a pretty good view of Vesuvius from the summit there.