There isn't a "main drag" where everything is located. Points of interest are all over the metro area, and a car will be needed to efficiently see things. Public transportation is not great - MAX bus system - and taxis are limited and dispatched by calling except at the airport. You will spend a small fortune in cab fare if you go this route. If you don't drive, how are you planning to get to Tuscaloosa? I don't think there is a bus system in Tuscaloosa, so a car will be a necessity there.
With that said, about the only area where you can reasonably stay and walk around to lots of things is Five Points South. The Hotel HIghlands is there and the Doubletree and Residence Inn aren't too far away. There are lots of restaurants, clubs and shopping in the area, but you will have to drive (or be driven) to other areas for seeing the sights.
Let us know more specifics on what you want to see and do, and we can help you a little more. If there is a way to re-think the no driving part, the trip will be much easier to plan and much more enjoyable.