I’ve done it a couple of times (well, from Manchester), very doable. I’ve wondered about the best ways of doing it. Perhaps not the shortest, and definitely not the quickest, but this is what I’d do:
Get to Edale. I don’t know the west side of the peak very well but there are loads of options. Up Jagger’s / Hope Cross / Beast (Or Potato Alley if sir prefers) / Rowlee climb / Lockerbrook. At Ladybower, I’d cross over then go up the steep stones to the top of Derwent Edge (For a bit extra go up to Lost Lad) then back towards the Ladybower Inn (Chose your descent). If it’s quiet, up the road and then along Stanage Top, but otherwise do the road climb up to Stanage Causeway.
Getting into Sheffield centre off road is a bit trickier. Porter Clough is well known but a bit dull. I’d prefer the techier Rivelin valley to Bole Hills (via Stanage Pole (RIP) to Redmires, but difficult to find without a guide.)
I think if you’re looking for the most efficient route then you’ll end up riding disappointing stuff. My route includes a few unnecessary / indirect climbs. Best to make a day of it and try and tie some great bits together.