Free: Contests & Raffles.
If you truly want a lightweight rifle, I'd suggest a 270 Winchester. Any cartridge bigger than that will have too much recoil. The 270 may be on the light side for elk, but if you use the right bullets it'll do just fine. Now if you're talking just an average weight rifle, like around 7 1/2 pounds, I'd say the 30/06 or either of the 300 Winchester Magnums would be hard to beat.
Most seem to be on the 300 band wagon. What about the .308? I have the savage.243 and love that rifle. I am thinking about the .308 savage 110.
If you are a great shot, then it really does not matter which cartridge. Take a rifle and shoot a target at 200 yards off hand. (no rest). For shorter distances, practice with a peep sight.If you are a terrible shot, then look at cartridges meant for elephants.