Generally, it's easier to get around in Africa during the dry season, which in east and southern Africa runs between late June and October. During that time, animals tend to congregate around water holes and rivers, so they're easier to spot. Plus, the foliage is less dense, so you can get a better view of them. One downside to the dry season, however, is that's also the part of year when lodges and camps are most crowded, and prices are often the highest.
In truth, though, the ideal time to go on a safari ultimately really depends on which parks and preserves you want to visit, what animals you hope to see, and what mode of local transportation you're interested in using. If you're primarily a bird watcher, for example, the rainy season is the best time to visit many areas. Serengeti National Park in Tanzania is the rainy season home for the vast herds of hoofed animals who migrate there in search water. So in that particular area, the rainy season is the best time to see the greatest abundance of wildlife, including the park's famous lions and other predators.
You should be forewarned, though, that if your ambition is to see a specific event, such as the wildebeest migration in the Serengeti, timing your visit can be tricky. Those events vary in date from year to year, based on climatic conditions, and are difficult to predict accurately in advance. So it's good to have a plan B in mind, as well [source: Fitzpatrick, et al].