I am a casual occasional recreational F-class shooter so be prepared to ignore me if a real expert comes along.
I don't know the alphabet soup on Remington model designations, but you need a long heavy barrel and a good stock. I have seen a number of Remingtons of that general description doing quite well.
Also, Savage makes a factory standard F-T/R (and F-Open) rifle that gets good reviews. I have a Savage that looks like that but is an earlier model with some gunsmithing.
There are others you could use, but you either get something that is not well supported with parts, accesories and gunsmithing, or you get something that runs the cost way up.
A .308 puts you in F-T/R where everybody is shooting .308 (or .223) so you are on a "level playing field." Any other caliber will shoot in F-Open versus heavier, more refined, more expensive rifles where the guys are paying for tighter groups and especially less windage.
You need a good scope. I use an 8.5-25X Leupold Target. There are others at higher and lower prices that I can't comment on except to say that you need a good clear scope with as wide a range of adjustment as you can get and with proper target adjustments. I was struggling yesterday with hunting rifle dials, mushy clicks, and poor tracking on a .22 yesterday and it was no fun.
If you will be or anticpate shooting at 1000 yards, you will likely need a scope base with some extra elevation built in. Very few scopes have enough internal adjustment to go from 100 to 1000 yards. Bases with 20 moa taper are readily available and will usually be enough. Most scopes will adjust from 100 to 600 yards, so if all you expect is midrange shooting, a flat base will be ok.
Harris bipods are very common. You can do better but you will pay a lot more for it.
Otherwise, you need a mat to go prone on and all the usual accessories for serious shooting. Check out the experienced shooters and see which gadgets are really used before you spend a lot.
Shooters are very helpful, but give them something to work with. When you go out for the first time, have your rifle zeroed dead on at 100 yards if you cannot do it at the range the match is being shot at. Use real match ammo, you need the accuracy and the ballistics are well known so the "come ups" for different ranges are pretty well standardized. Bought or handloaded, the 175 grain Sierra .308 is a known quantity and worth starting out with.
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