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While the 308 is one of the all-time target rounds and the .243 is very versatile, I would encourage you to split the difference and go with a 260 Remington or 7mm-08. The 260 shoots a .264" caliber bullet and there are is a good selection of bullets for long-range shooting available, in that size. The 7mm-08 is a .284" caliber so it will kick just a little more, but there is more factory ammo on the market for that cartridge. Either will shoot well out to 600 yards, with the right bullets, and neither kicks enough to be a problem. Both are excellent choices for target fun and hunting, if that might be in your future.
If you're going to do much shooting, especially at distance, you'll want to consider handloading, at some point. Any of the above will be easy to reload for and will provide the economy needed to send more shot downrange.
If you aren't a handloader stay away from the .260 Rem. it is a dying cartridge ! I heard that Remington isn't even building them anymore . That's a shame , it's a great little round . 6.5s in America come and go , always have and probably always will .
your best chance at affordable ammo if you don't reload would be the 308. i really really like th e7-08 myself (i have a couple) but based on your intentions and use,
i think it would be the more affordable choice. not to mention it will shine at those distances.
Most factory .308s come with a 22" barrel, which works well with it's moderate velocity. Adding a couple inches to 24 may give you 50 FPS more, but you might even find that kind of variation from identical rifles' slight differences. Most .308s come today with 1/10" twist rate, which works well with most bullet weights from 150-180gr. Your particular rifle may favor one weight (or brand) over another, but you'll only find that out after shooting a few different weights and types.
I'd also agree that the .308 is a very fine choice for your needs and is known to be a cartridge known for good accuracy. The other plus with a .308 is that it comes in a LOT of different rifles and models, maybe more than any other caliber. This gives you a huge range of types, models and most importantly prices for you to look at when choosing your rifle. If you're not worried about minimum weight, perhaps some heavy barreled models might even look good for your wishes. Generally speaking a heavier barrel will give you better accuracy, being stiffer and able to handle heat more effectively.
Either choice would be good, .243 is more varmint ready, .308 is the ultimate in versatility as it ballistically matches the 30-06 up to 180 grain bullets and seeing as how you can hunt anything in North America with the .30-06, you could do the same with a .308.
Except for a few of them, they will all put you in under budget with a very nice scope. Note that Remington prefers a slower twist in their .308s and will perform better with 150 grain ammo and lighter. The Rugers are 1-10" and will handle any bullet weight.
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