If you are looking to climb Mount Kilimanjaro, then the best airport to fly to is Kilimanjaro International Airport.
Flights from the UK regularly leave London, Manchester and Birmingham to Kilimanjaro (JRO) via Amsterdam with KLM.
An alternative is to fly into Nairobi and then take internal flights with Ethiopian, Air Tanzania, Air Kenya or Precision Air to Kilimanjaro International Airport. Unfortunately though, transfer times at Nairobi Airport can sometimes be quite long.
If you intend to spend time after your climb relaxing at Zanzibar, then you may be better off booking your return flight from Zanzibar Airport.