By Air
If you're going to climb Kilimanjaro or go on safari in Northern Tanzania, the most suitable airport to arrive at is Kilimanjaro International Airport (KIA). KLM has daily flights, Ethiopian and Kenya Airways also fly into KIA.
If you're going to visit the Zanzibar archipelago, or go on safari in southern or western Tanzania it is better to fly to Dar es Salaam. British Airways, KLM, South African and Swissair are the major carriers.
Regional flights to Dar es Salaam, Zanzibar and parts of northern Tanzania regularly fly from Kigali (Air Rwanda), Nairobi (Kenya Airways, Air Kenya) and Addis Ababa (Ethiopian Airlines). Air Tanzania has just gone bust and so does not fly.
>By Land
From Kenya: there are several bus services available between Tanzania and Kenya. Buses go from Mombasa to Dar es Salaam (about 12 hours), Nairobi to Dar es Salaam (about 13 hours), Nairobi to Arusha (about 5 hours). Moshi is also on the drop route for drop offs.
From Malawi: the crossing between Tanzania and Malawi is at the Songwe River Bridge. Direct buses between Dar es Salaam and Lilongwe depart several times a week and take about 27 hours. The alternative is to get to the border crossing and take a minibus (dalla dalla in Swahili) to the closest town -- Mbeya in Tanzania. Spend the night there and continue on the next day. Both towns have regular bus services to the rest of the country.
From Mozambique: The main border post is at Kilambo (Tanzania)from which you can get a minibus to Mtwara. To cross the border requires a trip across the Ruvuma River and depending on the tides and the season, this could be a simple quick canoe trip or an hour long ferry ride.
From Uganda: Daily buses travel from Kampala on to Dar (via Nairobi -- so you need to get a visa for Kenya). The bus trip takes at least 25 hours. A more manageable crossing is from Kampala to Bukoba (on the shores of Lake Victoria) which gets you to Tanzania in about 7 hours. Scandinavian also runs buses from Kampala to Moshi (via nairobi).
From Rwanda: Regional bus services run from Kigali to Dar es Salaam at least once a week, the trip takes about 36 hours and crosses into Uganda first. Shorter trips between the Tanzania/Rwanda border at Rusumo Falls are possible but the security situation fluctuates so inquire locally in Benako (Rwanda) or Mwanza (Tanzania).
From Zambia: Buses run a several times a week from Lusaka to Dar es Salaam (approx 30 hours) and from Lusaka to Mbeya (approx 16 hours).
By Train
From Zambia: A TAZARA runs from Kapiri Mposhi (Zambia) to Dar es Salaam. It takes two nights if you are lucky.
By Boat
From Malawi: The Ilala sometimes travels from Lake Malawi to Tanzania (Mbamba Bay) but schedules vary.
From Kenya (Mombasa): A ferry runs once a week from Mombasa, Kenya to Tanga. Fares are around $40.
From Zambia: A regular ferry connects Kigoma (Tanzania) with Mpulungu (Zambia). This ferry was the inspiration for the ' African Queen'. You can get off at smaller stops along the way, for example Lagosa, where you can stop to visit Mahale Mountains National Park.