Cars are already a lot more convenient than trains, and trains haven't gone out of business yet.
In some ways they are. Trains still have a few advantages (parking in urban areas, not having to drive, etc.). My point is that self-driving cars seem to kill off those last remaining advantages.
It's weird that California is investing so much in high speed rail. It would be much more forward thinking to invest in this California company's forward thinking, instead. http://www.google.com/selfdrivingcar/
avoidance of traffic is the biggest advantage trains currently (sometimes) have (and the first one dLew mentioned). As oil prices rise, it COULD get relatively cheaper than driving and flying. self-driving cars probably won't alleviate traffic much and will probably be more like uber as someone alluded to. hoping the hyperloop works out though.
I disagree strongly on the traffic point. That's the biggest potential advantage of self-driving cars. They will theoretically be able to network together to make traffic much more efficient and optimize routes in real time. It will basically be like everyone is riding in one collective conveyance, like a train, except that train can go a million different places at the same time. I think we're headed for something like what was envisioned in Minority Report.
Regarding speed over long distances. I think that, too, will ultimately be negligible once you factor in the travel at your destination. Besides, California isn't building a cross-country train. The main "high speed" line will be a few hundred miles.
The obvious huge downside of all of this is that for it to work, you can't have human drivers on the same roads rough ridin' everything up. Maybe there will be a proliferation of special driving tracks for the enthusiasts.