Tsavo West sits roughly halfway between Nairobi and Mombasa along the A109 highway. The main gate is Mtito Andei Gate, about 230km from Nairobi and 250km from Mombasa — a three to four hour drive from either city on the main road.
Two other gates give access to different parts of the park: Tsavo Gate on the eastern side, and Chyulu Gate on the northwest (useful if you're coming from Amboseli). A 4WD vehicle is strongly recommended for anywhere off the main park roads, especially in or after the rains.
There is no public transport into the park itself. Buses and matatus between Nairobi and Mombasa stop at Mtito Andei town, from where you'd need to arrange onward transport. Most independent travellers drive in with a hired vehicle, or join a tour from Nairobi or Mombasa.