The best time to go on safari is during during the dry season. You get to see more animals up close because the relatively few water holes mean trackers have a better idea of where to find the animals, and the thinner bush means better visibility.
During the summmer, there is lots of water so the animals disperse more and are less visible in the long grass and lush bush. In southern Africa, with four annual seasons, winter (July and August) is the dry season, but late May/June and September/early October are also fine times to go. In East Africa, which has a tropical climate, seasons simplify into the long rains and short rains. In Kenya, rainfall patterns vary from area to area, but east of the Rift Valley, for instance, the short-rain season is from November to December and the long rains March to May.
A popular time to go to Kenya's Masai Mara, west of the Rift Valley, or Tanzania's Serengeti - of which the Mara is the northern quarter - is during the migration. This is a long process - a huge cycle in which half a million wildebeest move up from the Serengeti to the plains of the Mara to calve - but the beasts usually start to move in March or April, arriving on the Mara in June or July and moving back in October or November.
I hope this helps.
Enjoy!