In reply to RedPin:
Hathersage would be a good place to aim for, especially if you're not hiring a car. Lots of major Gritstone Edges nearby, with Stanage being the closest, but you wouldn't want to walk up that hill every day so hitching is a good idea.
No 'hostels' that I know of, but probably bunkhouses, someone else might be able to help you there. There are B&Bs in Hathersage, that won't be a cheap option (£40 pn?). I guess you're not interested in camping.
Another option might be to see if some kind soul in Sheffield might put you up... you might get transport thrown in occasionally, otherwise the train is good and cheap. I'd have put you up, but I live in Canada now, lol!