There are several ways to get to Tanzania and they pretty much all go through Europe. Many people will choose to spend a day or two in London or Amsterdam on the way to their safari. In almost all cases the recommended flights will include an overnight flight to Europe and an all day flight south, over the Alps, the Mediterranean Sea and the Sahara Desert to Arusha's Kilimanjaro Airport.
Most flights from the United States will take you to either London or Amsterdam as a mid-trip location to change flights and direction of travel. If you are in Amsterdam on a KLM/Delta/Northeast flight you will be boarding a flight to Dar es Salaam. The stop in Arusha, where you will get off, is an in-transit stop. Be sure that your ticket and luggage are for Kilimanjaro Airport (JRO) in Arusha. After you disembark in Arusha the plane will continue on one hour to Dar es Salaam. When you depart Arusha after the safari you will board this same flight and head south to Dar and then return northward to Europe. The return leg is about three hours longer than the arrival flight.
There are flights through Nairobi, Kenya and from other African cities as well. But the Delta flight (previously Northeast or KLM) is the easiest to book and fits well into all itineraries. You will be met at the airport by your safari company and taken to your first night's lodging somewhere in or near Arusha. Most itineraries will have a whole day in the Arusha area so that travelers can sleep late, have a half-day outing, and recover from the impact of travel.
Just be sure that you are headed for JRO.