Wildlife in Tanzania : December-February and June-September but NOT late July-Sept for Serengeti safaris, the weather is relatively cool and dry, meaning less mosquitoes, wildlife is more concentrated due to limited water supplies, more visible due to shorter grass and you have more light to enable use of your new super-telephoto lens!
However, if you head for the Serengeti AFTER the mass migration to Kenya [July/August?] you're going to be a few animals short of a safari.
OK: November - December, short rains.
Worst: March-May due to long rains on the coast [and July-August for Serengeti safaris because wildlife takes a drought hike into Kenya

Where to go for Tanzania safaris: Tanzania tourism is often divided into either/or... Go North for teeming wildlife, grand scenery and tourist herds photo-grazing everywhere. The dream of a magical, remote safari experience may be killed stone dead if every time you get near something interesting - such as lions chowing down on a fresh wildebeest breakfast - a dozen open-top 4WDs screech up and jostle for pole-picture position. If you decide to go north, which you should, then travel with expectations of a wildlife Grand Prix, factoring in thousands of animals pursued by tens of jeeps and you won't be disappointed. Go South for less animals, less travellers, worse roads, better ambience, more cost.

A little advice on wildlife-safaris in Africa:
- Crowds tend to gather around the relatively rare predators; as far as other wildlife is concerned - elephants, giraffe, zebra, wildebeest, hyenas and so on - there is no shortage of these so tourists can get a lot more space and sense of isolation even in busy game parks like Masai Mara and Serengeti, with an experienced and knowledgeable guide.
- A good [human, as opposed to book!] guide is invaluable in any of East Africa's game parks, leading you to dramatic scenes and rarer animals before the herds of camera-wallies arrive, as well as providing information and lifestyle snippets of the beasts you come across. This is definitely worth paying a premium for!