I think it would happen this way: firstly you would automate things that go on a track - like trains - which I think we already have. Then secondly you could have certain lanes or roads that are designed for specific self-driving vehicles. This could help buses and trucks (in certain areas) get around. Then you would keep adding more of those, until many roads have those and very few manual lanes. It wouldn't immediately work in a big established city, but some existing highways or interstates can be fitted easily. Anywhere that has an HOV/Carpool/EV lane can be refitted to be an automated driving lane. Think of it like the Cash Only vs. Easy Pass toll lanes. At first there was 1-2 Easy Pass lanes, then the cash lane was just 1-2 - now we automate them reading license plates.
So to sum up, I think that the self driving will start in very specific areas, and then those specific areas would be expanded until they are the only areas. Especially if those roads contain automated taxi (Uber, Lyft, etc) which you can call with your mobile device. The main reason a lot of people drive is because public transportation in their area is junk and you have to go by it's schedule and route. But having a point-point self-driving option is way more convenient. It's like the subway, but it doesn't need tracks.
So to sum up, I think that the self driving will start in very specific areas, and then those specific areas would be expanded until they are the only areas. Especially if those roads contain automated taxi (Uber, Lyft, etc) which you can call with your mobile device. The main reason a lot of people drive is because public transportation in their area is junk and you have to go by it's schedule and route. But having a point-point self-driving option is way more convenient. It's like the subway, but it doesn't need tracks.