A guy doesn't have to spend a lot of money on a rifle just to kill a deer. If you're picky about precision, a little more money will get you a long ways.

The advantage in starting with a Savage 11/111 Trophy Hunter as a "budget friendly" rifle is the trigger. The barrel will be as accurate as most anything out there (and more accurate than most), the polymer stock will be as bad as anything out there, and the trigger will be better than almost anything else on the market. The trigger is really the deciding factor for me. Mossberg and Ruger have licensed the trigger design (as well as the barrel nut design), if you're interested in the Ruger American or Mossberg rifles. I'd favor the Savage.

I bought a Savage 11 Trophy Hunter in .22-250 for my dad a few years ago. Got $450 into the rifle, put a $100 Boyd's laminate stock on it, sold the package scope which came with the rifle for $50 and bought a $250 Bushnell Elite 3200 3-9x40mm, and Talley integrated mounts & bases for $35 on sale. I did the blocking & bedding work myself. Got all in for right around $800.

Similarly, I put together a Rem 700 .243win for my mom's boyfriend 2yrs ago - Rem ADL black friday deal at Walmart for $377 - $410 after tax. Boyd's stock, Leupold rings & bases, Nikon Buckmaster close out scope, self tuned trigger. All in for $760. (Thread for this one is in the Gunsmithing Projects Sub-Forum HERE

Either of these projects could have been done cheaper by keeping the factory tupperware stocks and simply stiffening the forend and bedding the action with epoxy. A length of All-thread bolt or aluminum angle, a router, and a Saturday afternoon and you'll have a solid stock for about $25.

The Savage won't need a trigger job, only an adjustment at most.

For cartridge, it's hard to say anything bad about the .30-06, other than it's incredibly boring. Effectively, it's exactly enough cartridge to kill anything in North America, with exactly as much recoil as most folks can tolerate in a proper sporter weight rifle without complaints and without padding or brakes, and ammunition is widely available for relatively low cost. It does exactly what's expected and needed of it. Anything else, largely, has some kind of "trade off" - as in the 7mm Rem mag is flatter shooting and more powerful, but also comes with more recoil and higher ammo cost, and lower barrel life. The 7-08rem has less recoil, but is less powerful for hunting larger game... But... the gold standard .30-06 is largely boring for many shooters. There's pleasure to be had in a simple ham and eggs breakfast, but it gets boring eating it every day. I'll never be without a .30-06, simply for the fact it will always do exactly what I need it to do, but in owning a lot of other rifles, it's not usually the first rifle I grab each season.

Looks like the Savage 111's are running about $525-550 these days, add $35 to that for bedding compound and a length of all thread to stiffen the forend, ditch the junky package scope and upgrade to a $250-350 Bushnell, Leupold, Nikon, or Vortex and you'll be in fine style.