In reply to sj1295:
There's plenty about within two hours of Newcastle, but by public transport it's a bit trickier! Northumberland-wise you can easily get to Crag Lough/Peel Crag (train to Haltwhistle or Hadrian's Wall bus to Twice Brewed), Simonside/Ravensheugh (Stagecoach bus to Thropton) and Kyloe (Arriva bus to Fenwick). There's obviously also Causey Quarry on your doorstep too.
Last time I checked, the North Yorkshire Moors were a public transport write-off from Newcastle, but that might have changed.