1000 yard shooting... most chamberings can do it, some just do it easier than others. if you aren't handloading, then the cost of ammo is a non-issue because match ammo for either chambering is expensive.
recoil is a non-issue as well, since you will be wanting a heavy barrel, and you'll want a good recoil pad w/ either chambering. by the time you get a big leupold, nightforce, or us optics scope on, a steel 20 moa rail and steel high rings, you are going to have a 14 pound rifle. throw another pound on for the harris bipod, and all of a sudden, recoil is minimized. in a sporter rifle w/ a poor recoil pad either one is a kicker.
so... rifle... remington 700 vls is a good way to get into long range shooting cheap. the laminated stock is very stable and inletted well.
and then, chambering options... the 6.5 mm bullets (so, cartridges like the 264 win mag, 260, 6.5x284, etc) will have much higher bc's than anything 30 cal. ditto 7mm (7 rem mag, 7-08, 7 rum, 284 win, 280 rem, etc). i really like the 7-08 and 6.5x284 for 1000 yard target shooting better than my 308.
if i were new at 1000 yard targets, i'd start from scratch w/ a remington 700 variant in 7-08, 260, or 308 - in that order. if i already had 1000 yard experience but wanted a factory available gun, it would be a cooper in 6.5x284.