As much as I love my 270 Win, since you are throwing elk into the picture I would second the 7mm Rem Mag. Nothing really wrong with the 300 Win Mag but as a first rifle it might be a bit much.
This is the perfect answer. There's nothing wrong with the .270, in fact it's my gun-of-choice, but a little on the light end for elk, and the .300 may be a little bit of a punch in the mouth for your first rifle. 7mm Rem Mag is a great choice - and, as always, throw the .30-06 into your decision-making process. It's always one of the answers to these questions.