And while I'm typing, this is what i'd recommend for a *new* TR shooter:
If you can, try out someone else's gear. Everyone says this, but it's not as easy as it sounds, so you're going to have to do some research and engage in a little trial and error. The goal is to minimize that expense with planning and research.
This is what will be required:
A bolt action rifle with a long barrel - approximately 30" is the norm. You need the barrel because velocity matters - F class is all about ballistics. There is no practical downside to a Remington or Savage action, especially when worked over by a competent smith. The customs are luxuries.
The chamber must be cut for appropriate bullets. For .223, nothing less than 80 grains, and preferably 90's. For .308, 185 Juggernauts are the minimum bullet I would use, and not a bad one to start out with. You will eventually want to try 200's of various kinds, but Juggs are pretty reliable until you're knocking on the High Master door.
Competent reloading skills. Nothing fancy here - you don't need a lab grade scale or fancy dies. Just good, basic practices will get it done with normal factory equipment. I got to High Master with a Chargemaster, in spite of many people telling me it couldn't be done. (not that good scales are bad - but they're certainly not necessary to compete.
A high magnification scope: the bare minimum would be a 36X Sightron or Weaver BR scope. They're 400 bucks, and will not hold you back. Second focal plane scopes with 1/8 minute clicks are what you want. Most find that 40x is about right.
Here's where it gets fuzzy:
Rear bag. Most shooters use an edgewood or equivalent. I have a minigater and It's great. Some use other bags. Some like to squeeze the bag for elevation adjustments, others like to move the rifle back/forward. If you're a bag squeezer, you need as softer bag, or even one of those tactical style bean bags. If you're a slider, a well packed flat bottomed edgewood is what you want.
Bipod. Most shooters use a rigid ski-type bipod. Weight matters here, as they vary quite a bit, but they all do basically the same thing. There are some with joystick controls, and others that are basically fixed. That's just a matter of preference and your positional requirements. Some shooters use what I would call a tactical style bipod, which is a broad category from Harris up to some giant expensive models. They will require a different shooting technique than the ski types. Again, this is a personal matter that is dictated by your position and preferences.
Trigger. A light trigger is a big help. You'll see lots of 2 oz benchrest triggers, and plenty in the roughly 1 pound range. A stock trigger is not ideal, but you can work with one if you practice good trigger control technique. A lighter trigger just lets you screw up more before you suffer any consequences. Generally, this is worth the money when starting out. Warning, though: 2oz is VERY light.
Stock. The stock chosen depends largely on the type of bag you want to use. for a bag squeezing technique, it matters less - you can use pretty much anything, but the tactical stocks are generally designed for this. For a traditional bag technique, you want a stock with a straight toe at a shallow angle. The McMillan XIT is a good example. My master class prone stock is not ideal, as it has a slight castoff and a steeper than needed toe. But I make it work.
The above will get you a rifle capable of winning any F T/R match. You can spend $6k or more on it, or you can go cheap and come in closer to $2000 (or even less if you buy used gear). What I would not do is skimp on any of the basic requirements - you need a good, long barrel that's properly chambered, and an appropriate scope on a gun shooting good bullets. That's 75-80 percent of it. After that, diminishing returns kick in HARD. You can go with nicer scopes, nicer stocks, fancier bipods, better scales, etc. It all helps, but only a little.
You can certainly shoot an F class match with less than this, but you need to keep in mind that the gun will be holding you back, and you won't get as much out of it or learn as much. If you are committed to F Class, get the right gear.
Best of luck. It's a lot of fun.