5r mil spec .308, you will like it. Or a 308 in varmint contour. Jmo Welcome to the forum. You are gonna get a lot of opinions, and some suggestion for a savage, even though you asked about rem. specifically. It's the nature of the beast.
No need to spend a lot of money...I have a Rem. 700 20" Tactical (@ $650) and it will shoot right at .5 MOA with factory ammo and between .2 and .3 MOA with hand loads. Replace the stock, good rings and bases and you are still in it less than $1000. Can't go wrong IMO