No love so far for a savage rimfire or hunting rifle? The mark ii in 22lr, 17hmr, or 22wmr is a great shooter. I hit a quarter the other day at 70 yards using cci velocitor's. Throw a good vortex scope on it, and you are good to go. But yes, learn to shoot off of a rimfire. a 17hmr is nice, but the ammo is more expensive than a 22lr; but cheaper than a .223 centerfire. You can stretch your supersonic range out if you practice with a 17hmr. It's good for small game as well. The savages come with the accutrigger and so you get a very light trigger pull, but do not compromise safety. You do need to practice with them though, because if you don't depress the blade on the trigger, and just pull the trigger, the rifle does not fire. You need to depress the blade and pull the trigger for the rifle to fire. Once you get the hang of it, and practice lots, it is no concern at all.
I bought a savage 300wm package gun for $450 and with reloads, I have gotten 1/2 inch and 1/4 inch groups at 100 yards off of a harris bipod lying prone. And that's with the cheap scope and rings it comes with. I took a moose with it using the federal version of the barnes tsx and that moose is still dead. Didn't make it far either: ran 20 yards and was down.
Another option is to go on tradex canada and buy one of the swedish mausers in 6.5x55 swede. At around the $300 mark, they are a steal of a deal. Very accurate and gives you more money to buy more ammo. The 6.5 calibers with 140 grain bullets and their monstrous ability to penetrate due to their high sectional density, you can take a range of big game. I would not take one for a bear, but that being said, I don't think that they wouldn't work on a bear, it's just I would rather take a 300wm for a bear (a 300wm to hunt a bear that is; bear defence the 12ga, and any of the centerfire calibers mentioned would stop a bear).
The calibers like .308, 6.5x55 swede, .270 are easier to shoot better as they are 'soft' recoiling. Less flinch development. But the tradeoff is less hitting power. The .30-06 is very versatile and can take any game on the continent. But there is more recoil and that can lead to flinch development; especially in a light hunting rifle.
To be able to shoot accurately, at some point consider reloading. Reloads tuned to a rifle make for great accuracy. If you do reload, Lee collet dies are cheap ($50) and they work just as well as my redding competition dies ($300).
Good luck!