As another ex-local I’d echo everyone’s suggestions. You can walk up the North Beach to Sewerby steps which come out next to the zoo and near the hall (tide dependent). The landtrain can take you back to town if tired of walking.
I’d add:
Danes Dyke is a nice place for a walk although its only a small wood, if you’ve seen Dad’s Army you might spot some of the filming locations.
Flamborough Headland is also a good place for a walk if the weather is clear, you can see right up to Robin Hoods bay on a good day. There’s some nice bays with loads of rockpools at North Landing and Thornwick Bay, spent many an hour scrambling around there as I grew up down the road. Just watch the tides though, they have a habit of quickly cutting off routes back despite the place you’re stood on being dry. There’s a few pieces of old ships and a couple of good caves to explore.
If it’s raining there’s an OK cinema in the arcades and more 2p slider machines than you can shake a stick at.