What James Drake has said is incorrect.
There are is a fairly large city bus system, as well as a light rail.
The system suffers from the same traffic that cars do. The city would be better served by add more routes and more light rail.
Austin is very low density, but high population. It's very difficult to serve this kind of area adequately with public transit.
Uber and Lyft have left Austin and have been replaces with a mismash of new rideshare apps with varying reliability and efficiency.