There’s a traffic free cycle route from the center to the airport, but it’s not exactly challenging. 10 mins up the A66 to Barningham is a good place to start off-road. Or look to the East, Guisbrough Woods, Wynyard & Carlton Bank.
Most of the local bridleways are patchy, short and or linear. There’s a small ridable loop near Newton Morrell, but just farm tracks.
As mentioned, best bet is get a road bike and head for Barnard Castle or Richmond Moors down the back roads.