In reply to abr1966:
Arrochar is an excellent suggestion, with the Cobbler, Ben Cruachan, Ben Lui, Ben Vorlich, the Black Mount all within easy reach, plenty of pubs, beautiful Loch Lomond and a sea Loch.
I'd also highly recommend Arran as an authentic Scottish island experience with minimal journey time.
Why not the Cairngorms? Aviemore is only 7 hours from the Peak and Perthshire maybe only five and a half. Much to see and do there.
Skye is 9 hours from the Peak, worth the effort but not a short journey.