In reply to The Norris: Lots of walking to be had.
- From Edale up onto Kinder Scout offers some challenging walking conditions.
- Most (many) of the Gritstone Edges have fairly easy paths along the tops together with good views and would combine with meeting you for a picnic/sunbathing at the base of the crag.
- Climb up from Edale onto Mam Tor and walking the ridge running East gives views over the Edale and Hope Valleys.
Castleton is OK if you like that kind of thing.
Personally it's a long time since I've been down any of the mines, but I think Treak Cliff was the most interesting.
You could also walk up from Castleton to Pevril Castle.
Explore the remains (not much to see) of the Iron Age Hillfort on Mam Tor.
Cycle hire at Fairholmes (Ladybower Reservoir).
Drive down to visit Chatsworth House/Gardens.