Nairobi's Jomo Kenyatta International Airport is the hub of East African air transport, and it has connections to many European and US cities. It's from these places that you're most likely to get a relatively cheap ticket, but it's worth checking out cheap charter flights to Mombasa from Europe too. Airport departure tax for international flights is USD$ 20.00 but is usually included in the cost of your ticket.

You can travel by bus or minibus shuttle between Kenya and Tanzania. The main routes are from Mombasa or Nairobi to Dar es Salaam and from Nairobi to Arusha and Moshi. There's also a once-weekly train connection between Voi in Kenya with Taveta (on the border), from which you can take local transport to Moshi. The ferry service that connected Mombasa with Zanzibar and Dar es Salaam is no longer running.

The main border crossing into Uganda is at Malaba, though Busia is an alternative if you're coming from Kisumu. Nairobi and the Ugandan capital Kampala are connected by road (the train no longer operates). Moyale is the main border crossing with Ethiopia, though those with serious 4WDs can also enter Kenya from Ethiopia via Fort Banya near Lake Turkana (this route isn't possible from Kenya to Ethiopia). There's no safe way you can enter or exit Kenya overland from Somalia or Sudan at present.