I'd echo the Tikka suggestions, but was going to say "Savage 10" as a budget option, Tikka Varmint (or Tikka CTR) if you have a little more $ to spend. When I last looked, the Tikka Varmints were around $1,300. In my recent experience (last 2-3 years) I've found that the Tikkas I've tried shot remarkably well with factory ammo. (2 Varmints, a CTR, and a T1x). The ammo chosen wasn't random though, mostly good stuff. (Hornady, Federal etc.-The T1x was very forgiving on a wide variety of 22 ammo, more than any 22 I've ever owned)
All of that to say-you may have a pile of surplus/bulk ammo that won't give you the kind of accuracy Tikkas can easily achieve, but if you're willing to get a good gun...it'll be in your safe once the lesser-stuff is used up.
Re: the Savage Axis (or Axis II?)...
I have experience with 2 X generation 1 Axis rifles, one in 223 and one in 22-250. In both cases, the guns were surprisingly accurate if you ran good ammo through them. Like Ruger Americans, they go "bang" for sure, and would probably serve most people fine. The lower purchase price is tied to allot of factors that may/may not appeal to the OP. Lower quality mags, heavy triggers, cheap metal finishing, lots of flex in the stock, etc. You don't need to know anything about guns to see/feel the difference between a Savage Axis and a Tikka T3x. It's whether or not the differences matter.
Just my opinion.