When cars started to take over from horse and cart, it still took 20+ years for them to become commonplace. It takes time for reliability to go up, and cost to come down. It takes time for the laws to catch up.
What seems like a blink in time to us now (1880s-1900) was 20 years for the people living it at the time. Cars were, at first, a freak sighting - then gradually, little by little, they caught on, and millions of horses were, ahem, retired.
A big step change isn't needed at all. Just gradual introduction and refinement until they are a compelling proposition.
As for the legal issue - the first generation of cars will require a human driver behind the wheel. They'll remain repsonsible for the foreseeable future. Likely the answer to long-term compensation for injured victims will be a shared insurance scheme - you pay the registration on your self-drive vehicle, and a portion of that goes into an insurance fund. If there is an accident and property/people are adversely affected, they get paid out from that fund according to a pre-determined formula ($x,000 for a limb on a 25 year old, $y,000,000 for a 45 paraplegic, etc). This isn't greatly different from schemes in place around the world today, where pinning fault on someone comes second to making sure those injured in traffic accidents are compensated + rehabilitated properly.