browning, being a relative new comer to long distance shooting, I was in the same boat that you are a couple of years ago.
You will get many opinions on caliber and brand of rifle to go with; but since you're on a tight budget, there are considerations to give some thought to and perhaps some research. You should consider things like barrel life, cost of reloading components and whether or not you are going to compete.
There is a reason why so many choose to go with a .308. Barrel life is very good at around 5000 rounds or so. Brass is cheap and bullets are reasonable. There are a number of competitions that are limited to the .308.
I don't hunt any longer and all I do is target shoot. Knowing what I know now, I don't think I'll ever buy another over the counter rifle. I look for a used custom or semi custom rifle, which will likely come in a nice stock that has already been bedded and a good barrel, such as a Rock, Krieger, Bartlein, etc.
As was mentioned above, it's all about the barrel. I know that a bunch of guys like the Savages because they can swap the barrels themselves, but that feature really doesn't do much for me. Most guys that I know that prefer Savage actions do away with the barrel nut anyway, in favor of a barrel with a nice thick breech end. In any case, Savage or Remington, get a good barrel.
A good barrel (as opposed to a stock barrel) will reward you with higher velocities (due to a tighter chamber), less copper fouling (due to a smoother bore), better accuracy as the barrel gets hot and more heft for a more stable platform (if you choose to go this route).
Heck, I sold a semi custom a little while back. It was built, by a pretty well known builder, around a Remington 700 action in a Bell & Carlson stock with a Timney trigger and Krieger 22" heavy varmint barrel. I sold it for $900.