Based on your list, I'd go with the Tikka every time. I believe it's certainly worth twice the price of a Ruger American and reiterate that I would avoid the American line if at all possible - along with other rifles in its price range. I simply don't think you'll regret buying a Tikka and can't say the same for the Ruger, Savage, Mossberg, etc. You could spend more on a Browning Hell's Canyon and possibly be happy with it, but I don't know that it would shoot any better and I personally find them unattractive.
Regarding cartridges, I would consider something a little more than the .243 if you can. I bought a couple boxes of that Hornady Custom Lite load in .308 Win for my son to use on his first deer hunt (he was 9 yrs old) and now it's the same load I use for whitetail hunting in TN. It shoots great and in my opinion is worth buying - although it's not cheap. I've tried that same Hornady offering in a 7mm-08, although I've not hunted with it. The Custom Lite is simply an accurate, low recoil load and I've killed 4-5 deer with it at ranges under 100 yds with excellent results. There's a Custom Lite load for the .30-'06, but I don't have any experience with it.
I would seriously consider the 6.5 Creedmoor. They tend to exhibit mild recoil with great accuracy. Maybe not enough for elk, but without having any elk hunting experience, I can't say. I don't think you'd go wrong with a Tikka T3 Lite in .243, 6.5 CM or .308. Maybe .30-'06, too - but as others have said, might be more than you want right now.
I'm a little suspect of the 7mm-08 in the Tikka because the one I had didn't shoot as well as I wanted, but others may have different experiences with it. It may help to check the rifle's barrel twist rate against the load you plan to use before you buy and search for user experiences with that combination. I think that was my problem with the Tikka 7mm-08, but I sold it before spending more time with it.
