Below is an overview of the Serengeti, month by month.  However, having lived and worked in the Serengeti for many years, our own favourite times of year would be:

February / March / April in the Southern Serengeti plains and the Loliondo and Gol Mountain areas to the east. This is wild country and with the onset of the rains, the plains are alive with birds and flowers as well as huge concentrations of plains game and a lot of predators.

August / September / October when the herds are ranging back & forth across the Mara river, particularly where it flows through the northern Serengeti, a beautifully varied area of plains, riverlines and kopjes (dramatic rocky outcrops) generally with very few visitors.