A lot of things will change. Because driverless cars don't require a driver, they will be operated continuously.
Instead of buying a car, most people will subscribe to a car service, probably for a number of miles a month, just like a cell phone plan. This will be much cheaper month-to-month than owning or having a car payment, although obviously it never gets paid off like a car loan. There will be upgraded tiers of luxury that can be purchased if you want to roll in a Benz instead of a Ford. Yes, we'll pay a premium for use during "peak" (rush) hours. We'll order a car using an app, similar to Uber but without the driver. Ride sharing will be common, but private rides will be available for a premium.
Edit: I wasn't clear as to why people would subscribe to a ride service rather than own their own cars. The reason is cost.
Driverless cars operated by a service will allow you to effectively split the cost of the car with everyone else who uses it when you aren't. Most cars now spend 95% of the time parked in a spot or driveway. There's no reason for a driverless car that can go to new passengers to ever not be in use, provided it isn't recharging or being serviced and there is someone close enough to be worth the trip who could be using it . Each passenger only needs to pay for his utilization of the car rather than for the whole thing if the car is shared. If you only spend 4% of your month in the car and could only pay for the 4% of the car that you actually use, that would be way cheaper than paying for 100% and only using it 4% of the time. Obviously demand for cars at peak commuting hours will complicate this, but sharing cars will dramatically reduce the cost of access to transportation.
Even people who spend their entire workday in a vehicle don't use them 24 hours a day. And, don't forget, most of those people are delivering something, and once driverless cars/trucks become the norm, those jobs won't be done by people anymore.
If you could pay $50-$100 a month--instead of a full car payment plus gas plus maintenance and repairs--for access to the same class of vehicle whenever you wanted, you would, even if it meant having to wait a few extra minutes for a pickup or scheduling the pickup in advance--which will be as hard as unlocking your phone and saying "Ok, Uber, I'm running eight minutes late this morning" and having a car pick you up at 7:38am instead of the usual M-F 7:30am pickup.
And driverless cars will be expensive to own. I seem to recall reading that Google's autonomous cars have about $250k worth of equipment in them. Obviously those are prototypes, and the cost will go down considerably. But a driverless car will remain much more expensive than a comparable dumbcar. Even if the cost of the computer and sensor hardware becomes negligible, the software still costs a lot to develop and maintain. And, don't forget, there will still be a huge liability assumed by the maker: if the car gets hacked or malfunctions and kills someone, they're getting sued. They will charge enough to make assuming that liability worth their while.
We'll pay that premium, though. We'll pay it to get our commute time back. We'll pay it to reduce accident risk to basically zero. We'll pay it to make sure nobody else ever dies because a drunk driver fell asleep at the wheel. And we'll make it a law that anybody else who wants to use public roads to go anywhere has to pay it, too. But it won't be much of a price to pay if we split it. Thus ride services.
Car makers will focus on making longer lasting cars engineered for continuous use. Except for the very high end, their market will shift from consumers to fleet models sold to car services.
Most auto body and repair shops will eventually close. A few larger chains will form by buying up other shops and compete for contracts to handle repairs for the large car services. The mechanics/body repair people who stay in those careers will end up working for them or directly for the car services in regional repair depots.
Gas stations will disappear, as the fleets will be electric, but that will happen even without driverless cars.
Driveways and garages will be conspicuously absent from newly constructed neighborhoods. Parking lots will mostly disappear, except for the few on which the car services will erect garages with inductive supercharging pads in the parking spaces. The cars will go there to recharge when necessary.
There will be more extensive drop-off/pickup areas at the entrances of shopping malls and office buildings for cars to pick up and drop off riders, though.
In general, we'll see the same things we've seen with disruptive technologies in the past. Entire industries, including auto repair, will be transformed, forced to pivot hard or be left behind. Many small businesses like auto shops will be eliminated or agglomerated into fewer, larger ones. Some, perhaps many, people will fail to adapt quickly enough and will be left behind. The trend has been toward a few huge national players emerging and using lobbying and regulation to entrench themselves, and, unfortunately, I expect that to happen here, too.
There's still the possibility of smaller car services, though. Car services will be capital-intensive businesses, naturally, but they won't necessarily require or have the massive infrastructure requirements, geographic monopolies, or regulatory barriers-to-entry (and resulting economic moats) which oligarchic cable, ISP, cellular, telecom, and power companies use to stifle competition.
It's very possible that we could see small entrepreneurs buying a few autonomous cars and offering their services through a ride brokering app/system. Smaller car services could also survive by targeting niche markets or providing boutique-style services. We might see serviced that specialize in cars equipped as a mobile office, or optimized for hi-def entertainment or gaming, or audiophile-aimed services with cars with really nice sound systems, or cars with built-in car seats for small children, or even specialized cars outfitted and targeted toward families taking long trips.
Smaller, local auto shops may be able to survive by catering to these smaller car services.
There will still likely be a small percentage of hobbyist car owners with privately-owned, old-fashioned driver-controlled cars, although they will eventually all be confined by law to closed tracks or private roads to protect the rest of us. (Both public opinion and the law will come to see drivers who insist on manual control, with its potential for human error, the same way they view those who insist on driving drunk.) Those hobbyists could also provide local auto shops with some business.
So yes, lots of people will lose their jobs. New jobs will be created. Life and progress will march on.
I think it's the small-town speedtrap-manning cops I'll enjoy seeing lose their jobs the most, though.