That rifle/scope combo is probably a Remington 710. It receives a few good reviews from users, and a lot of mediocre/poor reviews.

Based on your stated budget, I'd step up to a Remington Model 700 SPS, which will run in the $450 range, and double your current budget for a good scope. There are a lot of calibers that will fill your bill, and you've listed a bunch of good ones. With elk possibly on the ticket, I'd probably set a minimum caliber of .277 to .284. So, the .270 Win and the 7mm-08 will probably be your minimums. You can take a step up by choosing the .30-06 or the .308 Win. With any of these calibers, on elk, you'll need to keep their limitations in mind andbe selectivewith your shot choice and stay within reasonable range on elk. But, they're all more than adequate on deer at ranges beyond which I'd be willing to take a shot... I'm not much of a magnum fan, as I don't care much for recoil and muzzle blast. I'd rather have a standard caliber, and limit my shots to 300 yards, than pick a magnum that'll put them down at 400-500 but cause a flinch that will make me miss...