In reply to TobyA:
I think the Roman Road above Curbar is a footpath tbh
I rode Sustrans 67 from Stockport to Sheff on my Hybrid recently and my Cyclocross mate struggled on some sections especially around Woodhead.
I've heard about Cut Gate!
Ladybower circuit should be easy enough.
I did a circuit out around Clumber and it was great. Easy roads and good paths. Not very "Peak" though.
The Peak trails tend to be very busy and therefore slow. High Peak Trail, Tissington Trail, Monsal trail are all nice enough, but at the weekend are full of families and dogs.
I think there's a new Pennine Bridleway, that could be ok? Otherwise I'd be avoiding weekends, and maybe checking out some of the minor roads.
Tissington and high peak :
https://www.strava.com/activities/187887226
Clumber from Chesterfield (not a loop but easy enough to link to a circuit of the canal)
https://www.strava.com/activities/287826482
Sustrans 67 Stockport to Sheffield stations
https://www.strava.com/activities/325474829