Check this out.
http://www.boomershoot.org/general/TruthMilDots.htm
I know that my above answers were short and kurt, but your question really opens a can of worms. You'd be surprised at how many people there are that decide that they want to do some long range shooting and believe that all they need is a rifle with a heavy barrel and a really big scope. The fact of the matter is that as the distance to the target increases, it gets exponentially harder to hit the target (how much harder depends on the conditions). It takes practice, especially to hit a target at say 500+ yards with the first shot.
This is why I asked if you have a .22. A .22 is a very good place to start because shooting a .22 at 100 yards is a whole lot like shooting a .308 at 300. All you really NEED is a $200 Savage bolt action rifle, a decent scope with a good mount, some sort of rest, a brick of decent sub sonic ammo and time. If you get to a point where you can hit a golf ball at 100 yards just about everytime, off of a bipod you'll be doing pretty good. Then you can extend your range and try again.
Personally, I shoot a .308. It is a good round. Not ideal for 1000, but I haven't gotten there yet. The pros to .308 are these:
It is a capable round. Lots of guys shoot them out to 1000 with good success.
There is an almost infinite variety of components to build good ammunition and it is very easy to find a sweet load for a .308. I got my first one by just following the recipe in the Sierra manual. You can load for .308 for about .30 to .50 a round for very good match ammo. All you really NEED is a set of Lee dies, a good single stage press (I use a Lee Classic Cast), and a scale plus some other do dads. I would however suggest that you first get a good reloading manual and read it like a novel. I guess the Lyman book is pretty good.
Another advantage of the .308 is the barrel life. You should be able to get a good 5000 rounds through it before you see a noticeable decrease in accuracy.
If you really want to shoot centerfire on the cheap, the .223 is the way to go. Everything is cheap. The brass, bullets, primers, etc.