Keswick/Ambleside are the most central, the former has Whinlatter, Skiddaw and Borrowdale riding from your doorway, the latter has Iron Keld, Kentmare, Nan Bield, and a short hop to Grizedale and Claife.
Its a big place but you can get most places in an hour, give or take! As for digs, I'm not the best man to ask, as I never need to stay anywhere! I normally camp if I'm staying over somewhere. Are you looking at b&bs or camping/barns?