I would suggest the best off the shelf gun is a Savage, Cablela their store only and two models.
, a bit more tactical, shorter barrel (24 inch) and a rail, decent stock. Often has a $100 rebate and sometimes another $50 off. I think that get the price down to $460.
More a target varmint setup is the
That one has come in at under $300 with rebates. Stock a lower grade plastic than the 10T.
Both have the nice Savage varmint trigger (adjustable down to 1.5 lbs), both have what Savage calls a Varmint Barrel which is an almost bull barrel in profile.
With the heavy barrels you can target shoot a lot without the barrel going funny.
Lots of after market options for the Savages. Boyds makes some really good drop in stocks for under $150 (a bit of tang relief at the rear)
There are a lot of good field guns out there, not so much target guns. If its more target than field then weight is not an issue. 10T is more a split gun, the 12FV more the target or fixed varmint shooting .
I would consider 6.5 as more versatile overall. As good a bullet selection as the 30 caliber guns, a bit less overall costs for bullets, powder. Its more capable long distance do to the narrower bullet and better BC.
I do have one of the 12FVs (308, it fit in better with my other 30 caliber guns) that I have modified quite a bit (bull barrel and a Boyds stock, rail).
A plus in my view as I shoot a lot, is that I can replace barrels easily (and at home if you want to invest $150 or so in the tools)
Remington require a trigger to work decently.