A few of the answers touch on this, but I feel it deservers more focus.
You do not own a self-driving car, you subscribe to one, or rent one as needed.
The self driving car doesn't compete with the family car.
It progressively replaces the family car, with a different type of service.
Like NetFliks doesn't compete with broadcast TV, it replaces it with a different service (as compared to Channel 10 competing with Channel 9).
The self driving car service is much more like a Taxi.
There is no need to have a car sitting in the carport at night, when that car could be driving club goers into town. There is no need to have to have a large carport full of workers cars when they could be going to take people shopping.
The cost of a Taxi is mainly in man power. Thus self-driving taxis would be cheaper than Taxis.
Self driving taxi's are also cheaper than personally owned cars too.
Because they save on: the opportunity cost of not being used all the time (as they are), and on the parking costs at paid parking, and on the land costs in inner city household parking.
They will also have the advantage of never being out of fuel, since a self driving taxi (like any taxi) would not be sent to your location if it needed fuel or repairs. And it would always be properly serviced (unlike personal vehicles (I know my car is a year over due for a service, just cos I put it off)).
The do have some increased costs over personal cars, like the owner taking profits. So it depends on competition and supply/demand, the exact price point.
Thus Self driving Taxis will replace personal cars, and taxis.
The trick is waiting for legislation to catch-up.
While today's cars need supervision, the future ones will not.
We have been making progress in this, one of my professors worked on self driving cars decades ago, and apparently they were pretty good, but they were not allowed on the road at all. Todays are better, and are allowed on road with supervision. Tomorrows will be better still, and will be allowed on road without supervision.
Until then we can basically treat self-driving cars as just another convience/safety feature, like Electronic Breaking, and Cruise control -- nothing really changes.
Once legisliation catches up, then we have the real question.
The key difference between a self-driving car service, and a taxi is just the price.
In Australia, the UK and the US today, a Taxi might cost you about \$2.50/km. (I was surprised to find it so consistent.)
Lets say the price got down to \$0.80/km (My math suggests the Running/Replacement cost is about \$0.26/km, assuming 96,300miles/year,` \$0.16/mile for fuel, \$50,000 purchase cost, and replacing it after just 2 years).
There is another difference, in that before there is a person involved, where as now there is a machine. I suggest that on the whole it is irrelevant. In both cases any fear of the driver is statistically irrational. Some people are going to be scared of trusting a machine. Other people today are scared of leaving the kid with a stranger at all.
Now we can compare how people would interact with cheap taxis (ie self driving cars) to how the interact with taxis today.
If you google for "Send kids to school in Taxi" you will find a lot of result.
People do do it, it is quiet common. I imagine it would be come more common with a self-driving car.
Googling "Sending kid on playdate in taxi" returns far fewer relevant results. This forum discussion has people very unhappy with the idea of using Uber for such a thing. Though a lot of that comes down to the very unchecked nature of uber drivers.(The stranger concern is aggravated, compaired to for taxi's where the driver is going to lose their job if the suggestion of something untoward going on).
I suggest that with younger children, where the parents normally will sit and have coffee while the children play, there would be no change.
When we get to older children and tweens, I expect the cheep taxi service would come in to play. After all we already see children going to a friend's house via the bus or the other kids parent after school. So if the friend came to school via self-driving car then that would be the way they would go over after school.
In short, minor changes for younger children
For teenagers, today on a weekend job, they can't really afford to take a normal taxi. And for teens to young for work, on the pocket money they can squeeze out of their parents, similarly can not afford. What they can afford is public trainsport.
Now even the cheap-self driving taxis are still significantly more expensive than today's public transport. I thus suggest that a lot of teens would be fairly unchanged, with regard to their travel. Change may be that instead of walking to the trainstation, they get a pick-up.
But it is still out of their reach to do things like take a self-driving car for a day trip to another city etc.
On the other hand, in places that have no public transport system (like where I grew up), there may well be more significant changes. Parents may be more willing to pay for a self-driven taxi, than to drive them themselves. Particularly since the parent may not own a car.
In short the additional freedom doesn't amount to much.
It is still too expensive, though it is close.