Without knowing your shooting experience with rifles, but having read your statement about "feeling the rush," there are plenty of calibers that will cause you to feel physical pain, for up to days, following your shooting. Since you're new to hunting, I'm guessing the large recoil guns you have experience with and enjoy are handguns, maybe wheel guns like the .454 Casull or .460 Magnum, or .500 S&W. I've shot wheel guns with over 8" barrels and it's different than getting pounded in the shoulder.
Part of hunting is making an accurate shot for a clean, quick, and humane kill for the animal's sake as well as your ability to recover the animal for the meat or your trophy. If you're varmint/predator hunting, you don't need anything that produces massive recoil.
If you're hunting for the sake of acquiring meat, you also want to use a sufficient caliber to do the job, but not so excessive that you destroy the meat.
$3,000 will get you plenty of rifle to hunt deer with a nice scope and a reloading set up.
Where do you hunt in Minnesota? When I lived there, rifles were no go in the southern part of the state. Shotguns only when I lived in Kandiyohi County.
Think about what you want to accomplish with the rifle and ensure it gets that job done before thinking about the amount of recoil.
There's plenty to read using Google about the benefits of handloading.
.260 Remington is a great caliber for hunting. It's not going to give you ridiculous recoil, quite the opposite, but most folks who shoot matches, or anything for that matter, go for what gets the job done with the least amount of recoil. .260 is becoming, if not already, one of the most popular cartridges for competition and it's fantastic hunting option, especially for deer. The 6.5x55 is nearly identical and Europeans hunt just about everything with it. 260 will carry more velocity and energy at 1,000 yards than .308, and drop and drift less on top of that.
What range do you plan on shooting to? What distance do you have the ability to practice regularly?
One more thing, usually the more recoil a cartridge produces, the more expensive it is to shoot based on the size of the components, volume of powder, etc it uses. Just another thought.
