Miami is a car city. The distances are large, public transport coverage is patchy, and the weather makes walking any distance uncomfortable for much of the year. If you can rent a car, do — but be prepared for aggressive drivers and expensive parking, especially on South Beach.
That said, the Metrorail elevated train connects the airport to downtown, Coconut Grove, Coral Gables, and points south. The free Metromover loops around downtown and Brickell, which is useful if you are staying in that area. Metrobus covers the rest, though routes can be slow and infrequent.
For getting between South Beach and the mainland, rideshares are the easiest option. The causeways connect the Beach to downtown, but traffic on the MacArthur and Julia Tuttle causeways can be terrible during rush hour and on weekend nights.
Tri-Rail is a commuter rail line running north through Broward and Palm Beach counties — useful if you are heading to Fort Lauderdale or flew into FLL.
Citi Bike operates bike-share stations across Miami Beach, downtown, Wynwood, and Coconut Grove. The flat terrain makes cycling practical, though bike lanes are limited and drivers are not always attentive.