Jacksonville is a car city. Its area (875 square miles — larger than London) and low density make public transport impractical for most journeys. Rent a car unless you are planning to stay in one neighbourhood.
The exceptions are downtown and the Riverwalk, which are walkable, and the water taxi that crosses the St. Johns between the Northbank and Southbank (runs Thursday through Sunday, check current schedule). The Skyway, an automated monorail that connects a few blocks of downtown with the Southbank, is more of a curiosity than a practical transit option.
Driving in Jacksonville is straightforward once you understand the geography: the St. Johns River curves through the city from west to east, and most of the interesting neighbourhoods are on its banks. Riverside and Avondale are 10 minutes west of downtown; San Marco is 10 minutes south via the Main Street Bridge. The beaches are 30–40 minutes east on Beach Boulevard (US 90) or J. Turner Butler Boulevard (SR 202). The JTA operates bus routes throughout the city but service is infrequent on most routes.