It is best to fly into Newcastle International Airport (NCL). NCL isn’t a particularly large airport but it connects well to London, Amsterdam, Paris CDG (several flights per day to each). From NCL, there's multiple options to get to Durham (see below for details).
While we recommend flying into Newcastle, it is possible to get to Durham from any other UK airport. The two closest options, other than Newcastle, are Manchester or Edinburgh. It is also possible to use one of the big London airports such as Heathrow, Gatwick or Luton. These are all much bigger airports and more likely to have direct flights.
From any UK airport it is relatively easy to get to Durham on the National Rail, where there will be regular trains (likely every ~1 hr). We would recommend pre-booking your train tickets and seat reservations from sites such as the trainline . Typical times are ~3 hrs from Manchester, ~2 hrs from Edinburgh (including a ~30 min bus from the airport to the city centre) and ~4 hrs from any of the London airports.
From the airport, you can get to Durham via taxi, public transportation (e.g., metro, train, bus), or rideshare (uber). From Durham train station, it is about a 25 minute walk to Grey College and Calman Learning Centre.