Rotorua is not a walkable city in the usual sense — the attractions are spread over a large area. A rental car gives you the most flexibility, especially for reaching Wai-O-Tapu, the Waimangu Valley, and the Redwoods forest, which have no useful public transport.
The central sights (Government Gardens, Polynesian Spa, Ohinemutu, the lakefront) are all within walking distance of each other. For Te Puia and Whakarewarewa Village, it's a 25-minute walk from the centre or a short taxi ride. Many accommodation providers run shuttles to the main geothermal parks.
Cycling is possible but the main roads are busy. The lakefront path and some of the residential streets are pleasant; the Redwoods mountain biking trails are excellent for off-road riding.