Kenya has about 250 airports and airstrips (of vastly varying quality) and plenty of airlines connecting Nairobi with Mombasa, Kisumu, Nanyuki, Malindi, Lamu and the national parks/reserves of Amboseli, Masai Mara and Samburu. While many flights are heavily booked, flying around Kenya and its neighbouring countries is a relatively safe and fairly cheap way to cover a lot of ground. The Kenyan train between Nairobi and Mombasa is also a popular form of transport, despite the fact that the rolling stock, tracks and other essential works have been allowed to deteriorate. The trains are considerably safer than travelling by bus or matatu on the same route, but they are less frequent and take much longer.
Kenya has a network of regular buses, matatus (usually minibuses), shared-taxis and private taxis. Matatus were notorious for their involvement in horror road smashes (the Mombasa-Nairobi road in particular), but the new government seems to have tamed them with the forced installation of seatbelts and speed governors. Robberies by ambush (mainly in and near Nairobi) are still a problem, but not as common as they used to be since police now search most matatus before they can leave Nairobi. To further reduce the risk of both robbery or accident, never travel by matatu at night. Cycling is best done in rural areas due to the chaotic traffic on the main roads, but the distances between towns and the poor condition of roads needs to be kept in mind.
If you're bringing your own vehicle to Kenya you should get a free three-month permit at the border on entry, as long as you have a valid carnet de passage for it. Keep in mind there are certain routes in northeast Kenya where bandits roam and you must obtain police permission before setting out that way. Hiring a vehicle in Kenya (or at least the national parks) is a relatively expensive way to see the country but it does give you freedom of movement and is sometimes the only way of getting to the more remote reaches. The condition of Kenyan roads vary considerably.
For the more maritimely minded, sailing on a dhow along the East African coast is one of Kenya's most worthwhile and memorable travel experiences. Some of the most popular, and most expensive, dhow cruises leave from Nyali on the mainland opposite Mombasa Island. For a cheaper dhow trip, go to Lamu.