The transition to driverless cars is (IMHO inevitable. At some point it will be cheap enough that the additional cost will pale in comparison to the lives that will be saved as well as the simple convenience of being able to do something else while commuting somewhere.
Likewise I see this kind of thing replacing many forms of public transportation. There will simply be a fleet of cars. You'll say where you want to go and some system will route people and cars to destinations.
But, the transition won't be quick or easy. You need look no further than the aviation industry to see why.
Basically, automation in modern aircraft is a double-edged sword. It seems to erode the ability of pilots to actually fly [1], software errors causing deaths [2] and (I can't find the link to this) I also read a study that in more automated planes, pilots are more likely to believe erroneous instruments rather than their own senses and experience.
The issue won't be how the car normally behaves because as demonstrations have shown, current systems require very little human intervention.
The issue will be extraordinary circumstances plus the huge liability problem of any errors.
Example: if someone runs a red light and causes a crash, killing someone, that person is responsible. If an automated car does the same thing, the manufacturer will be responsible.
That alone will impede adoption.
Instead I think you'll have what we already have: slowly adding automation to cars. Cars already have radars and can stop themselves from colliding, they can park themselves and so on.
But at some point the driver will need to go away and that will be a tremendously challenging leap forward for society.
[1]: http://www.tourismandaviation.com/news-4530--Pilot_Reliance_...