At sea level, the .308 will get you to around 800 yards on deer and 500 yards on elk, depending on the ballistics of your chosen hunting load. At a conservative elk hunting altitude of 5000 feet, you can tack on another 100 yards or so to the above figures.
Those ranges are based on a 185g Berger hunting hybrid @ 2650 and take into account ballpark energy minimums of 1500 ft/lbs for elk and 1000 ft/lbs for deer. Another thing to consider is the velocity needed for bullet expansion on impact with your chosen bullet. That may have an effect on your range limitations as well.
Michael Eichele has written some very good articles on this site regarding the .308 and its capabilities. I encourage you to read his writings and draw your own conclusions.
Many of the experienced long range elk hunters on this site (I am not one of them) will tell you that legit long range elk rifles start at .300 Winchester Magnum and go up from there. Some will argue for the fast 7mm's as the beginning. In any case, a true long range elk rifle is not a beginner's rifle. Because of prior shooting experience, you may not consider yourself a beginner. Relative to long range hunting, however, you most assuredly are a beginner.
That's not to say that you can't start bigger. You can go bigger and be successful at it if you are able to dedicate the time and money it will take. I recommended the .308 as a starting point with emphasis on trigger time and reps in mind and with the idea that you could take the same rifle afield and hunt deer and larger game with it at appropriate ranges.
You can go bigger, but understand that the cost per trigger pull will escalate quickly, even if you reload. If you decide to go big, I would skip the F-Class route and spend the money to attend a long range hunting course like those offered by Shawn Carlock. That will shorten your learning curve and make every cent you put into practice more effective and efficient.
In the end, the gear you choose is just the start. The real difficulty and expense involves tuning the nut behind the trigger.
In closing, I will refer you to this thread:
Cold Bore Challenge
It gives a person a pretty good idea of the standard of marksmanship that a long range hunter is truly chasing. It is worth keeping in mind as you select your gear and choose your path.