Winnar!
Srsly. I hate to be the bearer of bad news, but you aren't going to just buy a gun capable of hitting small things at long distances and just go out and do it. If I were in your position, I'd buy something a bit cheaper first to get the basics out of the way.
As was mentioned:
-Get yourself an accurized 10/22 or the like, or my personal small caliber favorite, a Savage 93R17, and work on your short game first. They're small, but I promise you'll have a blast (no pun intended) with it. My Savage 93 is fucking surgical. 22LR is cheap to shoot all day, and 17HMR isn't too bad either.
-10/22 (22LR) is good to roughly 100yd.
-93R17 (17HMR) is good to atleast 150yd., possibly 200 on a windless day.
-Once you've had your fill of the short game, move on to your mid game to start practicing how to adjust for distance, wind, temp., pressure, etc...
I'm talking around 400-600yd. Distances won't be so far that you'd have to be exact with your math, but you can atleast start getting the feel for what adjustments do what.
.223 would be a great round for this. Ammo price is on the rise, but still isn't too bad, especially since you won't be blowing through mag after mag since you're doing precision work. I'd personally pick a Remmy 700 or comparable Savage for this type of stuff (though I'd pick a bolt gun for any precision work) over say, an AR-15.
-After you've got the hang of that, I'd move onto a .308 for your 600-1000yd. stuff. Again, ammo is getting expensive but you really don't burn through a whole lot with a bolt gun. I've got a Savage 10FCP in .308, but a Remmy 700 or similar would do the job just as well.
Your math calculations as well as your technique will need to be pretty damn good at this point as a minor fuck-up will turn into a big fuck-up by the time the bullet has traveled 1000yd.
-.300WinMag would be my choice for 1000-1,250ish yard.
-.338Lapua would be my choice for 1500ish yard.
-.408CheyTac or .50BMG would be my choice for anything past that.
You'd better bring your A-game for shots like that, though.
Rule of thumb for precision guns: Expect to spend around as much for an optics setup as you spent on your gun itself. Shit ain't cheap.
Ammo isn't cheap either. Unless you're a prodigy or some shit like that, you're probably going to have to shoot A LOT to get good. Hence, why I recommend starting out with the smaller calibers and working your way up. Unless you're baller (and you might be), you'll shoot yourself broke trying to learn the Kung Fu of distance and precision shooting. Once you get into the .308+ realm, you'll be spending about $2-6 everytime you pull the trigger...