Here's the scope I was mentioning earlier, and it's right at the price point you were looking for. Sniper's Hide reviewed this scope some time ago, and there's a link to that video review in the page I'm sending you. The scope served me well on my precision rifle for 3 years, and is now doing nice work atop one of my .22's. It has a MIL based reticle with MIL based turrets (it's nice to have these matching), and it has adjustable "tactical style" knobs, as opposed to the hunting style you're probably most accustomed to. The glass is really nice for the price point, and I never had any real issues with the scope.
The one drawback I can think of is the fact that it is a 1-inch tube, which tends to limit your adjustment range when compared to 30mm and 34mm tubes (but, honestly, I don't know of a decent $300 scope that's available in anything other than a 1-inch tube). Regardless, a run of the mill 20moa canted base should still get you to 1,000 yards with this scope:
http://www.midwayusa.com/product/59...mm-1-10-mil-adjustments-mil-dot-reticle-matte
Again, you may want to look into the Tikka line of rifles, given your requirements. They're the budget version of a Sako rifle, and are made in the same facility and on the same equipment as the Sakos. Mine has a detachable magazine, is damn accurate for the price, and does great double duty as a hunting rifle. I don't know if the "Scout Compact Tactical Rifle" that I have is still being produced, but it was virtually identical to their line of "Varmint" rifles I've still seen in stores (I think the only difference is that they screwed a Picatinny rail onto mine before shipping it to the store -- you could do that yourself if you wanted, and get the canted base right away).
I am 15 years old, 6 foot 1, and 240 lbs(mostly muscle) I don't think recoil is a problem.
Seriously though, think about some of the suggestions you're getting around here on the recoil, even despite your size. It seems like I've shot damn near every cartridge out there, from light plinkers to punishing lightweight safari rifles. Anyone can take recoil for a couple of shots, and it isn't necessarily just an issue of sheer mass or toughness. Just figure that if you're looking at playing in the precision long-range game, you'll be spending a LOT of time proned out behind a rifle shooting shot after shot. It can wear on you.
For instance, I always considered a .308 to be a light recoiling gun. But, I've left the range with a bruised shoulder on a number of occasions after shooting my .308 precision rifle. I can't remember my total weight with the new scope/mount (it's definitely heavier), but I was originally running a gun that came dressed out right around 9lbs with a scope, bipod, sling, and magazine. After a long range session shooting 175grain bullets it sure didn't feel like the .308 was as light on the recoil as I always regarded it.
Heavy magnum calibers can really beat you up. They're great for some applications, but as others have already said, they can cause you to develop bad habits... even if you are big/tough enough to take a hit (and I'm sure you are -- I felt invincible at your age, personally).
One last (and in my mind very important) consideration for you as a new shooter: the big magnums and screaming fast rounds typically seem to burn barrels a lot faster than some of the more mild/classic cartridges. If you want to get good you'll need to shoot a LOT (as I'm sure you already know from shotgunning). The .308 is a versatile cartridge that tends to give very long barrel life. The ultra magnums tend to be on the opposite end of the spectrum.