The historic center of Angra is compact and flat enough to cover entirely on foot. From the main square you can reach the cathedral, the main convents, the harbor, and the garden in under fifteen minutes in any direction. The Monte Brasil trails are a longer walk but the path from the edge of the city is clear.
For the rest of Terceira island, you need a car. The volcanic interior calderas, the Algar do Carvão lava tube cave, the Biscoitos wine region on the north coast, and the various villages where the traditional street bullfighting takes place are spread around the island and are not served by practical public transport. Rental agencies have offices at the airport; it's worth booking in advance in July and August when supply is limited. The island is small enough — roughly 30km east to west — that you can cover most of it in a day of driving even with stops.