A car is essentially required for Augusta beyond the immediate downtown. The city is spread out, Sconyers Bar-B-Que is a drive from the centre, the Canal visitor centre is several miles upstream, and there is no meaningful public transit network for visitors. If you fly into AGS, rent a car at the airport.
Downtown is the exception. The Riverwalk, the Museum of History, the Woodrow Wilson Home, and the Broad Street bar district are all within comfortable walking distance of each other. If your hotel is downtown, you can spend a day without a car.
Uber and Lyft are available and reasonably reliable in Augusta, though wait times can be longer than in larger cities. During Masters week, surge pricing can be extreme and availability unpredictable — a rental car is strongly recommended that week.
Cycling is pleasant along the Canal towpath and sections of the Riverwalk. Bike share is limited; bringing or renting a bicycle for the Canal trail specifically is worth considering if that's a priority.