I'd wager you will continue to see a mix of transportation modes at first, then a proliferation of non-vehicular transport for shorter/medium range trips, with greatest acceleration of change occurring in the highest person-density areas like NYC, LA, Chicago, and other big cities with their eyes on ending the scourges of public parking, local emissions, traffic accidents, and just generally the unsightliness of a car-centric world.
While the ride-hailing self-driving cars will be a great boon to those not wanting to drive, saving parking and potentially preventing accidents, I do believe SDC's will come in waves starting from the SW USA and California before spreading to the rest of the country/world. In these areas, public transit will likely be less useful, since for instance in Tucson, AZ, the city is massively spread out and busses would simply incur a much greater time-cost due to their non-personalized route.
Moving forward, as SDC's become more % of the fleet in urban/suburban areas (still won't be in rural areas for another 5-10 years or more), you will probably see more variety of SDC's. Instead of having a 50 person bus, you can have a 5-7 person SD shuttle/minivan (think Waymo Chrsyler Pacifica) that will take 5 people with similar routes to the same place.
Once SDC is highly penetrant in terms of %, I'd expect a great prolfieration of electric scooters, electric pedal-assist bikes, regular bikes, pedestrians, and all sorts of non-vehicular transport. I myself would ride a bike to work if not for the very low-skill drivers I encounter daily in my commute. This happens to be in an urban area with very poor bike infrastructure and very large hills. SDCs + electric bikes would solve these two issues, IMO, and they are probably ~10-15 years out in my location.