There are two traditional rain seasons, the long rains fall during the months of April and May and short rains fall between October and November.

The Serengeti plains are +/-2000m above sea level with a temperature ranging from10 to 20 degrees Celsius. During mid winter the evenings and mornings are often chilly, the Ngorongoro Crater at +- 2200 m above sea level could drop to below freezing during June / July.

The closer to the coast you get the hotter and the more humid with Zanzibar experiencing almost unbearable humidity during the summer months, so please bear this in mind when planning your beach holiday and choose a lodge offering fans and air conditioning.
Tanzania

Highlights and best times to visit the Serengeti National Park:
February–April:     Migration on the southern Serengeti plains

August–October:  Migration crossing the Mara River

July–January:      Good predator sightings

June-July:           Migration through the Western corridor

June–March:         Best time for general game viewing

April-May:            Cheapest time to travel (and still really good)

The Serengeti has a lot more to offer other than the migration, not only would there be less visitors, the sightings of other plains game would generally be better and the big cats are more prolific outside of the migration months too.

The excess supply of food leads to the cats becoming lethargic (as they lie and digest their latest feast throughout the day) and you generally do not see them that often except early morning and or late afternoon.  The cats would also be outside of their normal structures and or territories because finding food is easy with the migration passing through.

Highlights and best times to visit the Ngorongoro Crater
Ngorongoro Crater has good game viewing all year round because of the resident game and the density of the animals within the Crater.

February-March: Lake Ndutu in the greater Ngorongoro Crater Conservation witness the migration pass through
July and August: can be very dusty and busy – this is also the most expensive time to visit Ngorongoro Crater.
April and May: Due to the rainfall and are beautiful, everything is in flower and you actually see more because the soda lake is full condensing the animals into the space available.

Highlights and best times to visit Tarangire National Park
Tarangire is arguably Tanzania’s second best wildlife destination offering massive elephant herds and wonderful Baobab trees. Being a dry season park the best time to visit would be from June – November, traditionally it is not visited from March to the end May.

The rest of the parks and reserves within the north of the country would be good to visit all year round; this includes Lake Manyara National Park, Arusha National Park, Loliondo game controlled area and Grumeti Reserves.

Mount Kilimanjaro National Park, (climbing Kilimanjaro)
The best time to summit the world’s highest free standing mountain would traditionally be July – October, but it is possible to climb Kilimanjaro anytime of the year because most of your climb would be above the clouds so you will not be affected by rain, but to avoid a muddy first 2 days then best do not climb during April / May and November / December.

Another tip would be to try and coincide your climb during a full moon; this makes your summit attempt not only easier but more scenic as this last part of the climb takes place during darkness so to enable you to reach the summit for at sunrise.