I cycle this area frequently.
I would recommend the road through Edale - it is one of my favourites - quiet and scenic. Jacobs ladder at the western end up to Mam Tor is a nice climb.
Personaly, if I were traveling east from Edale to Norfolk Park, I would steer clear of the A57 east of Ladybower. From Hope (bottom of Edale) I would go via Hathersage, and the A6187. Half way up the hill from Hathersage, bear left and climb past Cam Height - much nicer road than the A57. Then it is a long fast descent into Sheffield along the Ringinglow road. At the bottom go left at the T on the Eccleshall road (A625). About 1k further down the hill there are some traffic lights, keep straight on onto Psalter Lane (do not follow the Eccleshall road/ down to the left). Keep going straight on at the next lights to the small roundabout, take 2nd left into Cemetery Road. Down Cemetry road to the bottom, right and then left into Boston Street. right and left onto Sheldon Street. Across Bramhall lane into Clough Road and then down to cross the Queens Road at the bottom onto Duchess Road and you will be on the edge of Norfolk Park.
HERE is a map of the route if you are interested.
You can follow the A6187 all the way up to surprise view and to the A625 if you want but I think the smaller road described is far nicer and much less busy.
FWIW, the Eccleshall road, Psalter lane section was used in the Tour of Britain a couple of years ago.