You barely have to leave the city for your first taste of the bush. Nairobi National Park sits just 7 km from downtown, the only major wildlife reserve inside a capital anywhere in the world — lions, rhinos, and giraffes on open plains with the skyline behind them. A half-day morning drive before an afternoon flight is entirely feasible.
Further afield, several excellent full-day trips are possible. Lake Nakuru National Park (about 2 hours north-west) is known for its flamingos and is one of the best places in Kenya to see both black and white rhino. Lake Naivasha (1.5 hours) offers boat trips to see hippos and a wide variety of birds from the water's edge. Amboseli National Park (4 hours south) gives close views of Kilimanjaro and large elephant herds — better as an overnight stay, but feasible in a long day with an early start.
The Hell's Gate National Park near Naivasha lets you cycle through gorges and geothermal landscape — unusual for an African park in that you can do it without a vehicle. Ol Doinyo Sabuk National Park, one hour from the city, is a quick and undervisited escape into forested hills with buffalo and colobus monkeys.