From Cockermouth there are some very nice areas. Head South and you are in Borrowdale, with Whinlatter, Newlands and Honister passes and on the other side of those passes is Keswick.
To the North East you can climb up Uldale onto Caldbeck Common which is a nice open moor.
North West you can head up the Coast to Allonby and Silloth and return through the relative flatness of the Solway Plain.