I'm not a long-range Bubba myself, 500m is my current furthest, but I'll quote the standard line given to the question:
"By the time you're good enough to shoot 1000m, you'll know enough about rifles to know exactly what you want."
Hopefully someone will come along soon with a suggested starter rifle, but since I'm here now...
Were it me, I'd get a Savage 110 in .223 with a good-quality scope. You could plink with milsurp 5.56mm and buy affordable varmint/target loads for other purposes. Recoil would be minimal, and bore erosion not bad. There are plenty of folks using tricked-out AR-15s for 1000m competition, and every year thousands of Marine recruits learn to hit a man-sized target at 500m using beater M16A2s, military ammo, and iron sights. So a Savage 110 w/ good scope in .223 should be an extremely affordable option to hone your skills.
That's assuming you already have a .22LR. If you don't have a .22LR, the standard advice is to get one and shoot the holy heck out of it for a year or two, out to as far as you can shoot. Ammo costs 2c a shot, no excuse not to send hundreds or rounds downrange every trip. There's a ton of great buzz on the board about the CZ 452 rifle, which seems to be the hands-down favorite target .22LR for its price.
Run a search (I describe the process in my above post) on either of the above rifles to get more info. And don't skimp too much on the scope!
Now, floor is open to people with far more knowledge than I.
-MV