Jackson is a car city. Public transport (JATRAN buses) exists but runs infrequently and is mostly used for commuting; visitors will struggle with it. Walking works only within downtown and within Fondren, but not between them — the two-mile gap along North State Street is unpleasant on foot.
Rent a car at the airport, or use rideshare for hops between the museum district, Fondren, and dinner. Distances are small and traffic is light by American standards. Street parking downtown is straightforward and metered; museum garages are inexpensive.