You can have a fantastic Tanzania safari at any time of year!
During the “long rains” of April and May, and the “short rains” of November accommodations are cheaper, lodges are less full, and there are less other cars in the national parks. It can get a little difficult if you want to travel to really remote areas during rainy season, however the roads between Arusha, Lake Manyara, Ngorongoro and Serengeti are good and well maintained, and still very much passable even during the rains.
The rest of the year is generally dry, after the rain season you have high and green grass, lush landscapes.

The mass migration of wildebeest moves from Kenya into the Serengeti in December, January, February and March. This is a very popular time to travel to Tanzania, and can be a spectacular time to visit, with the herds generally attracting a lot of predators and some real wildlife drama!
Note that the river crossings of the wildebeest take place around July or August, however, this is notoriously difficult to predict. It is really just a matter of taking your chances, booking in advance and seeing if you get lucky enough to witness this sight in person!
July, August, September and October are usually very busy (particularly July and August due to school holidays). These months are all dry, mild and, as always, great for wildlife viewing!