Torif - given your intentions, the first question I would ask is whether you reload your own ammunition. Although it is not absolutely "essential" to get started in F-Class, eventually you will want to roll your own if you want to compete at a high level. There are two reasons I mention this. The first is that you will want to consider the cost of reloading equipment and supplies at some point, which will be above and beyond the cost of a complete rifle setup. Secondly, your intentions and physical description would match up very well with using .223 Rem as a caliber, which would put you into the F-Target Rifle (F-TR) subdivision of F-Class. Unfortunately, most .223 Rem offerings in commercial ammunition are not at all ideal for shooting at 600+ yd, due to the lighter bullet weights used. In order to get started (if you don't already reload), the .308 Win would be a better choice for F-TR shooting at 600+ yd, as a few heavier bullet options are commercially available. However, the .308 Win will have
substantially more recoil than a .223 Rem, which might be a deal breaker in terms of just getting started. These are things to think about.
Given your location, the Milan Rifle Club might be a good place to get started:
www.milanrifle.club
Last I heard, they host High Power/F-Class matches out to 600 yd and are located a bit closer to you than Des Moines, being just across the river almost straight south of Davenport. There are certainly other options in the general region, but most will be in the range of 4+ hr driving time. I shoot F-TR at Eastern Nebraska Gun Club (
https://engc.us/home), slightly southwest of Omaha, but that would also be a bit of a drive for you. Nonetheless, you are more than welcome to come out and shoot with us next season. In fact, I have a couple .223 Rem F-TR rifles that I shoot regularly in matches in addition to .308 Win. I would be willing to load up rounds and let you shoot one in a match to see what you think. I'm sure you would also find plenty of discussion with our shooters regarding an F-Class rifle setup and competing in F-Class. We have a few shooters from IA that may also be able to point you to a venue closer to home. I appreciate that the logistics of traveling to Omaha are not as appealing as finding a range that hosts F-Class matches closer to home, but the offer is open if you ever want to make the drive. You can always contact me via PM here if interested, or just for answers to any questions you might have.
To elaborate a bit more on the .223 Rem, most F-TR competitors that use one are hand-loading bullets in the 80-95 gr weight range, as these bullets are noticeably more resistant to wind deflection than the lighter (77 gr and below) bullets typically found in commercial ammunition. Because the bullets most often used are much longer (heavier) than those found in commercial ammunition, you want a rifle with a chamber that has a long throat better able to accommodate them. Barrel lengths of 30" (or more) are also the norm. These long bullets also require a relatively fast rifling twist rate (7.0-twist is generally sufficient) in order to fully stabilize them. One rifle that satisfies most of these requirements that also falls well within your price range is the Savage 12 F/TR rifle in .223 Rem.
https://www.savagearms.com/content?p=firearms&a=product_summary&s=18154
The one caveat to that is that I don't know exactly how long the throat of the Savage 12 F/TR rifle is out of the box is, but the barrel length and twist rate are spot on. Further, the throat length can be extended with a special tool. There are other considerations with the Savage rifles, those being the Accu-trigger, which some don't care for, and that the factory barrels are generally pretty rough inside the bore. Nonetheless, factory Savage rifles can shoot straight out of the box, and I have seen many very fine scores posted by folks using them. Further, once the factory barrel is shot out, it can be replaced with a quality aftermarket barrel chambered any way you like. So another attractive feature about the Savage rifles for those just starting out in F-Class is that they will typically burn out the factory barrel while learning more about F-Class shooting and gaining experience in the discipline. By the time the factory barrel is burned out, they will be more experienced and ready to upgrade with a new barrel. For that reason, the factory barrel really shouldn't be a deal-breaker, and as I mentioned, factory Savage rifles can shoot extremely well. Again, something to think about.
Best of luck with whatever you choose to do in competition precision rifle shooting! It's not exactly the "least expensive" sport you could have chosen, but you'll meet a lot of great folks and enjoy the thrill of competition.