In reply to Rich Payne: Welcome to the wonderful world of SED and Mcr Uni!
Froggatt, Stanage and Bamford are all accessible from the Manchester-Sheffield line but all involve a ball-busting bike ride up steep hills.
None of the quality Lancashire quarries (eg Wiltons, Anglezarke) are particularly accessible by public transport, though I haven't actually tried.
Hobson Moor and New Mills Torrs are your best bet for genuinely accessible evening crags, though in reality you will need a bike for the former (nearest station is Broadbottom probably). As mentioned the Chew valley (particularly Den Lane) is accessible from the Leeds line.
Better is to have a partner who will pick you up from Hathersage for the East or Buxton for the Roaches and Peak Limestone. I actually do this quite often as have to share a car and it takes some of the stress out of getting out of Manchester.
But frankly, public transport and climbing don't mix. Better to hook up with someone or get some wheels. If you want to get out sometime drop me a line, I'm occasionally heading straight out from the uni with the car.