The Hobbit tells the adventure of Bilbo Baggins, sixty years before The Lord of the Rings. The three films were shot in New Zealand, and some places appear in both sagas. Start at the Hobbiton Movie Set: it was rebuilt for these films, and on a guided tour you walk up to Bilbo's round green door.
On the North Island, the Mangaotaki Rocks near Piopio are where Bilbo gets his small sword Sting, and the slopes of Tūroa on Mount Ruapehu hide the secret door of the Lonely Mountain. In Wellington, the Wētā Cave shows props made for the films. On the South Island, the Pelorus River is the river of the barrel escape, and the valley of Paradise near Glenorchy is the land of Beorn, the man who turns into a bear.
Check before you go: Hobbiton and the Mangaotaki Rocks are on private farms and open only on a tour, and Earnslaw Burn is a long mountain walk. You need a car to move between places.