Hamilton is a car-dependent city and a rental car is by far the most practical way to explore both the city and the surrounding region. The city centre is compact enough to walk, but the Hamilton Gardens and the zoo are more easily reached by car or taxi.
Waikato Regional Council runs a city bus network (Busit!) that covers the main suburbs and connects to the university. Services are frequent during peak hours but limited on evenings and weekends. Taxis and Uber are available. For day trips to Waitomo, Hobbiton, or the Coromandel, a rental car is effectively essential unless you join an organised tour.