IIRC a military range is 600+yardish so probably around 300 meters would be nice, I have little experience and I'm really just looking to be able to go to a range sit down, take my time and make some clean shots... I think that should be my approximate range but I'm looking for any guidance i can get.
I don't remember anything about you...so are you in the military or a veteran, or do you have any shooting experience?
In competitive shooting circles, 600 yards is midrange.
You can't just buy a rifle, scope and bipod and make 600 yard shots on paper. Well you can, but it will cost much more than $1000.
That being said, some of the most accurate rifles out of the box are Tikka T3s and Savage Model 114 Classics (any savage with an accu-trigger). The Tikka T3 comes in .308, link below.
700s are good, but overrated and expensive. If you are just paper punching then why get a large caliber? 22-250 and .223 are highly regarded as great long range calibers for shots on paper, but don't have enough energy at that range for many hunting applications.
Also, to get good groups (consistently) at ranges like that you normally need really good hand loads. Good glass is important and expensive, but the most important factors are a great trigger, a good bedded barrel/stock, and a correctly floated barrel.
Check out Ruger No. 1's, don't rule out single shots!
Basically don't get your hopes up unless you know what you are doing...and if you do know what you're doing then get a bad ass set up and post pictures!
Link: T3
https://www.budsgunshop.com/catalog/product_info.php/products_id/23570