Tanzania takes pride in its famous snowcapped Mount Kilimanjaro (5896 m), highest summit of Africa. There is a wide choice of hiking trails leading up to the 4000 m high north and south circuit paths around the mountain's main dome Kibo. Three routes lead from there to the Uhuru summit. Climbing Mt. Kilimanjaro is not particularly difficult technically, but can be dangerous due to high altitude and cold.
On Mt Kilimanjaro independent trekking is not permitted, so you will have to join an organized (group or private) trek at all times. There are huts on only one of the routes up Mt Kilimanjaro, the Marangu route, so if you take an alternative route you will camp out.
But hiking Tanzania doesn't stop at Mt. Kilimanjaro. Second in height is Mount Meru (4566 m) an active volcano in nearby Arusha National Park. Independent trekking with guide is a good option here. Other areas for hiking are the Crater Highlands in Ngorongoro Conservation Area (NCA), the isolated Mount Hanang (3417 m), the Usambara Mountains and the densely forested Udzungwa Mountains.
Best time for hiking in Tanzania is during the dry months, from December to February and from June to September. However (heavy) rain can fall throughout the year. Trekking in Tanzania means you must at least hire a local guide and bring camping gear and supplies.