I think it's probably going to end up like flying. Nearly all of us are quite happy to be flown around the place by aircraft which are pretty much under constant computer control. Some people go to the time, effort, and expense of getting a private pilot's licence and doing it themselves, but they are very much in the minority.
I'd be perfectly happy with an autonomous car for the routine stuff, especially when the technology gets to the point of not requiring any driver input at all. It will be like having one's own personal chauffeur-driven car, but without having to worry about the chauffeur wanting the evening off.
