Speaking as a southerner who has now been in Durham for 15 years and has, to my horror, started pronouncing 'pass' and 'last' (and 'mask'!) to rhyme with 'at', I can confirm that there is a lot to get used to, weather- and otherwise. Tow Law is always the go-to spot for snow. The snow also makes it harder for the charvas there to nick your wheels...
Like OON, I've only ever done Hartside in the car. Cycling Durham City's hills is tough enough!
My aim for the winter is to do the Cow Green Reservoir to High Cup Nick walk with snow. Not sure it's the most sensible thing I could do, but it's such a great walk that must look amazing in mid-winter. Might continue to Dufton, which must be close to you, OON.