Plenty of buses connect Nairobi with Mombasa and pass close to the parks, but as you'll need to rely on expensive transfers and game drives from the lodges to move beyond the main road, you'll rarely save any money by travelling this way if you plan to explore the region and it will usually work out cheaper to rent a car (or join a group) travelling from Nairobi or Mombasa.
Car & Motorcycle
The gateway to Kenya's southeast is the truck-choked Nairobi–Mombasa highway. The only other paved road connects Emali (the turn-off to Amboseli) and Oloitokitok, a border post with Tanzania. Elsewhere, you may be fine in a 2WD but a 4WD is recommended, especially in the rainy season. Most travellers rent a vehicle with driver in Nairobi or Mombasa.
With the new train service between Nairobi and Mombasa beginning in mid-2017, the stops in Voi, Mtito Andei and possibly Emali may prove useful (or, at the very least, preferable to the road journey). Whether the new service will be followed by new possibilities to rent vehicles in the two towns was unclear at the time of research.