Self-driving is great for people who:
- Cannot drive: elderly, small children, sight or otherwise impaired persons, sick persons.
- Shouldn't drive: drunk, high or emotional
- Don't want to drive
- Like to replace driving time with other times (reading, calling, sleeping, ..)
It isn't for people who:
- Want to drive themselves
- Live in remote areas with limited means: too expensive
- Like to own cars
If you think self-driving cars it's better to think of a universal bus system. Only with smaller buses and no driver. On-call, always available in most denser areas. With time delay or own-your-own in more remote areas. In fact, the bus system will become obsolete. Most regional trains too, but they'll probably stay since the infrastructure is built already.
Accidents will drop by a factor 10, the car manufacturer will most likely self-insure. Millions of lives will be saved, tens of millions injuries prevented. Broken cars or cars in trouble will call roadside assistance. Parking space inside cities will be freed up, possibly removed. Big quick-charge stations near power plants will likely be built if it's electric (and it will be). These stations will likely have a service center embedded. Many (most?) people will no longer own their car, but it will be a long while before a family with children gives up their station with all their family stuff in there. If the price is right, they'll still own it.
Regarding timeline and state of the art: still a ways out for full autonomy (no steering wheel), but the race is on. Most expect 2030 - 2040 to be the tipping point. Closed-off sites will be first: mining sites, predictable routes (intra-airport), .. Big problems are snow, rain etc .. expect the self-driving fleet to be temporarily unavailable when conditions are poor. Later it will be better.
Multi-tiered and application specific fleets will emerge: shared rides (think bus), luxury rides, school pickup/dropoff rides (with extra safety features). Just like today. I'm not counting on reduced congestion though. If a service becomes cheaper and easier to use, demand usually skyrockets ..