All of the rifle and caliber suggestions given here are perfect ... for the guys who use them, but they are not you. Everyone tends to believe their setup is the best. Here's my advice, but it won't be about specific caliber or rifle.
Since you are new to big game and elk hunting, I would recommend that you refrain from getting any of those calibers that hurt in both directions and/or tend to weigh a lot. Even big guys are influenced by heavy recoiling rifles, and it's much better to shoot straight than hit harder. Also, even a horse hunt requires lots of walking, so a rig that weighs a lot might not be the best way to go. Get a rifle that fits YOU and that you feel confident about shooting. I have had no trouble killing elk with the .270 Win,. 280 Rem and 30/06. And, although I wouldn't recommend it, the biggest dead bull I ever saw was killed with a .243 Win using a Barnes bullet, and it dropped in its tracks. So, be sure you use a well-constructed bullet. Good luck on your hunt!