Baton Rouge is a car city. The major attractions are spread across a large area — the downtown capitol district, the LSU campus a couple of miles south, and various plantation sites further out — and public transport connections between them are limited.
By car is the standard way to get around. Parking is cheap and usually available downtown and near the LSU campus. The highway network (I-10, I-110, US-190) can get congested during morning and evening rush hours and on LSU football Saturdays, when the whole city gridlocks around the stadium.
The Capitol Area Transit System (CATS) runs buses throughout the city, but schedules are limited and the network is primarily designed for commuters rather than visitors. Route planning through the CATS website or Google Maps is recommended if you plan to use it.
On foot, the downtown historic district between the New State Capitol and the riverfront is walkable — 15 to 20 minutes from one end to the other. The LSU campus is also pleasant on foot. Between these two areas you will need a vehicle or a rideshare.