If you would like to shoot F-Class, then you have two options F-TR (Target Rifle) which limits you to either .308 or .223. The gun can be up to ~18.2lb, use a bypod. The other option is F-Open. In this class you can shoot any caliber and use a front rest. There seems to be two popular calibers although the field is wide open, .284 Win and 6MM (with tons of variations on this). In my opinion, the .284 Win seems to be used by many of the National Champions.
Cost wise, F-TR is more affordable to get started. to start, bipods are much cheaper than the front rests. Generally $200-$500 depending on type compared to the front rests than can easily run $1000 and more. The rifles will run you about $2000 give or take for a competitive rifle and I would highly recommend starting with a used gun which could help you save even more and see if you like it.
Scopes will run you between $1000 & $2500. Again, you can always start with used as many competitors are constantly buying and selling equipment there are a lot of good quality glass out there to be had. You will want something adjustable up to 50 power, 30mm tube.
In addition you will need a mat to lay on, a rear bag and of course reloading equipment if you want to really compete. Sadly, store bought ammo will not be competitive and will only frustrate you. Again this equipment can be purchased used, but does not often come at much of a savings. You don't need fancy to start, a good single stage press, competitive dies, a good quality scale, etc.
On the scale, don't bother buying cheap $100-$200 digital scales. They just don't have the accuracy or repeatability that is required. Again, you can find used. I got my Sartarius which new sells for well over $3000 for $250 at an auction for a lab. You can start with a good beam scale such as a Haus 10-10 which can be very accurate.
I started with a standard rifle and over the years have spent more than if I had just purchased something used and got started and would have most likely been money ahead if I had. But you can start off with some of the basics.
If you are interested, check for ranges near you that have F-Class matches and go and watch a match. Talk with those there about getting started and they can help you get started. Many competitors will even loan you equipment to get started. A great way for you to see if you want to get really involved in this insanity. Through these networks, you can also find people to help you learn how to reload for competition, find good used equipment and generally make as few mistakes as possible.
Good luck and enjoy. You will find that the majority of competitors are very friendly, more than willing to help someone just starting out.