As some one who recently took a taxi from Newcastle Airport, I can tell you that you would certainly fit a lot of luggage in the boot/back of the cab. It can easily fit what you require. Also, there is a taxi rank just outside the arrivals lounge after you come through customs etc. This is naturally a 24hr service and there is a kinda "marshall" bloke who radios certain taxis to come if you need a larger one and none are available at the rank.
However, bear in mind that this is going to cost you the thick end of £60 to get to Durham. Ok, that's maybe a bit too much, perhaps £50, but I doubt any less.
You could always get the Metro (like an underground/tube thing) from the Airport to Newcastle Central Station, then, having gotten off the metro, get the lift (2 trips perhaps) up to the railway station, and if you phone ahead (will PM you telephone number if you want it), they will get someone to meet you to help with your luggage. then you could always take the train to Durham (about £7 in total per person so far), and get a taxi from Durham station to college.