You will certainly need a car to get between Edinburgh and the Lake District. It is a substantial drive to Lakes from EDI - about 2-3 hours or so as I recollect, depending on which area you want to visit. The roads in the Lake District are extremely narrow and winding, so you want to make sure you are up to that sort of driving, and that you have the time. Things that look close together on the map can actually be a fairly substantial drive. It is worth it, as it is a magnificent area.