I also have the savage axis, got it for $350 at Dicks. Came with a scope (not a good one). Shoots just fine, 3 in. 5-shot groups at 100 with corelokt ammo off a crappy rest. Sure it'll do sub 1.5" groups if ya find the right loads or handload it. Real nice synthetic stock and matte finish for such a cheap gun. I bought it just to have as a beater for bad weather days but I'd have no problem carrying it regulary.
The downside: trigger is stiff and the bolt isn't the smoothest. That seems to be where they're cutting cost.
I would look at the Marlins. I have one in .06 and one in .270. The .06 shoots m.o.a. at 100 yards with big green's ammo and plan on loading some 165 bllistic tiips for it shortly. Haven't fired the .270 yet but guy who sold it said it was a shooter.
A used Remington 700 ADL or a used Ruger 77 can certainly be had for around $500. Chances are you'll get rings and a cheapo scope on it too for $500. I'd steer clear of bottom end rifles at the price range you're in. Shop and be patient and you'll end up with a good rifle.
You can't go wrong with Savage if you want a good shooting gun out of the box.
My uncle is a metallurgist and does a lot of testing for Savage and told us they have some the the most rigorous specs as well of all the gun manufacturers that have components tested with his company.