Durham is about 20 miles south of Newcastle. I've never taken a taxi there, but it would probably set you back in the region of £50 (I'm guessing here) as it would have to return empty. Lansdowne (I spelled it wrong the first time ) might give you a quote.

There are fast and frequent trains between Newcastle Central Station and Durham (around 4 trains per hour, 15 minutes journey, £5 each way). A cab from Gosforth to Central Station would again be around £8-£10.

Train times at National Express East Coast.

If you're at the Metro Centre ("Metrocentre Gateshead" on the above website), there is a train station attached to the shopping centre, but you'll still have to change trains in Newcastle. There's also a direct bus to Durham - the number 44 - but bear in mind that this is a local bus, and it takes 90 minutes to cover 20 miles .