I don't think there are any Direct trains from the Airport but the Dark Peak has a train service from Manchester Piccadilly.
This could put you staying in Hathersage or Edale both of which have a walks from the door (Edale more strenuous ones, but all easier than a lot of the Lakes). Hathersage is a bigger 'town' and you can get the train to Edale for some variety, in fact you can get a train to all of the hope valley.
Dark Peak would give you Kinder Scout, Mam Tor, the great ridge from edale. You have Stanage Edge, Fox Houses from Hathersage.
If you're looking South peak district then Buses seems more likely. I'm sure other will have recommendations on getting around and a good base to start from.
Lee from Doncaster does his walks using public transport so you can get a big choice without driving (which makes the pub post walk pint more enjoyable)