We were in Fulford, lived in and around york for the last 35 (blimey) years.
Yes, some parts of Tang Hall are still less desirable, as are parts of Acomb and Clifton. I can highly recommend fulford, good prices, just over the millenium bridge from ‘Bishy Road’ (middle class heaven if you like that kind of thing), no real rough parts, nice walk up the river in the town. BUT its not easy to get over to the north from the south unless you cycle.
Decide if you want to be in a village or the city, maybe post a bit more about what’s important.
Villages to the west get more expensive as they’re commuting distance to Leeds as well as York. The east is generally a bit cheaper but the A1079 can be bad for traffic. A166 is generally better. Haxby/Wiggington/Strensal are all nice places and cycling commute distance.
If you’re in the city it’s a fair way to any hills even for road cycling, the closest is probably Terrington bank and the Hambleton hills to the north.
As above, traffic is really really bad, particularly on the ring road on the north side of the city (clifton moor area) at usual commuting times. There’s plenty of park and rides, but they don’t all have bus lane routes into the city. Cycling is the best way to get in.