The consensus so far seems to be that self-driving cars will see us to the Megacity Of The Future, but I'm going to buildoff @Mikey's Short-Term worry, because I frankly think that makes much more sense. And I have reason to believe so.
I live in Houston.
Unlike just about every other major city in America which was built during the 1800s in compact little clusters (necessitated by the average walking speed and distance of your average American at the time, because that was how you got around. That or horses.), Houston was built up after the era of the Interstate, and it shows.
HOUSTON IS YUUUGE.
The thing is, Houston grew up with a pre-existing network of Interstates (10 and 35), many US Highways, and a sprawling grid of subsidiary highways. Why buy a 10 foot box downtown when the same cash could buy anyone in Houston a two bedroom house just outside town?
This trend just kept continuing; Houston now has THREE HIGHWAY LOOPS, each loop, like the ring of a tree, showing the size of the city at various ages. We've upgraded another highway to Interstate specs (US 59 is now I-69), and most spoke highways are being expanded.
US 290, a main feeder road into Houston, was one of the first places to be sub-sub-suburbed; traffic is a nasty snarl that never seems to ever abate. It's rush hour all day; the HOV lane shuts down once a month due to congestion. They're doubling the lanes of the highway, but by the time they're done it's gonna be at capacity again.
Because, in general, people want space. They want a yard, and a fence, it's The American Dream.
This is not to say that there aren't serious city-folk who love New York. We have a few here; they all live in the Galleria. They live in a world of foot traffic and high rises. We have public transit as well; it's fares are constantly sucked out to build highway infrastructure. But Public Transit doesn't scale out well over vast distances, which end up leading to that nasty gridlock.
And this is why self-driving cars will enable spread out sprawl, as opposed to centralized cities!
Self-driving cars deliver us cake, and help us eat it as well. Point to point travel is always more efficient to the rider in time and energy expense. The reason the busses don't reach out to my area is because there would be too many stops necessary to build the type of hub-and-spoke system they depend on to be efficient. But Self-Driving cars gives me the convenience of a bus ride (cheap, shared with others) but with the efficiency of owning a car (leave when I want, direct travel).
Uber and Lyft and Google will, of course, do what they can to encourage people to live closer to where they work to save money (I mean, save the Environment.) on maintenance and fuel, but that's just an added expense. I already pay extra in wasted fuel, wasted time, car maintenance, etc. I'd continue paying a premium to live out where I am, because I want breathing room. And as self-driving cars take over more and more of the bandwidth on roads, I see more and more people able to afford the "premium" to live out in the sub-sub-suburbs, because the cost will be spread over everyone.
The rich will own their own self driving cars to save the inconvenience of having to use an app and wait. Many people will utilize self driving cars to get where they need to go. Busses will still flourish where they can run at capacity, but many "almost always empty" lines will be replaced by automated mini-busses. Job schedules will become more flexible, as you can give your RideID to your boss, and they can see you're actually on your way. People will be able to spend more time "socially networking," even if they aren't actually sitting next to each other. Self driving fleets, able to optimize highway traffic, will double, triple, or more the amount of people able to travel on a highway, and the increased speed of computation power will make high speeds possible. EVERYONE will be the cool guy doing 120 on the highway. My 35 minute commute will go down to 10, and I'll still be able to live outside the city.
City centers, as hubs of commerce and business, however, will still be your huge shining cities on the hill. The expectation is that those huge, shiny towers are residences full of people; I see it more likely to be just like Houston; huge business blocks and commerce buildings. The high efficiency of the automated autobahns will allow more business to be concentrated at the centers, where transactions can be handled faster and inter-business shipping becomes more efficient. Additionally, shipping bandwidth into and out of a city will be much wider, without the rush hour inefficiencies requiring businesses to spread out from the main hub (I even see companies like FedEx running "priority trucks" that pay extra just to be able to pass other traffic, even during high-volume times).
You're still getting your huge, fancy cities with shiny towers, but they're not jammed full of people like some kind of Dystopian future; they're 24 hour hubs of commerce, and their happy and healthy employees will be able to use the Cars Of The Future to spread out, each to their own preferred residential density, using highly efficient point to point traffic to reclaim the hours of life we all lose.
I look forward to it!