I will ask that you please refrain from using the term SNIPER. They are highly trained specilized fighters used in the military and police forces.
I am not one for long range hunting. If you are to lazy to stalk or wait till your pray comes into a reasonable range you don't need to be hunting in my eyes. I am mainly a varmint hunters and shoot ground hog I can pull shots off had out to 200+ yards Last summer my hit rate was 95%+ and I would not attempt to shoot deer at 300 yards. My longest ground hog shot was 250 yards. I have red misted Ground squirrels in CA at 300+ yards. That is just my view. There is a forum just for long range hunting LongRangeHunting.com
You are new to the world of rifles correct. You need to realize that shooting at 1k yards is a big chore and takes some knowledge. It is simply not buy rifle go to range and shoot 1k yards and be good at it. You need to know a lot of things.
The only way your getting into the 1k game with only $3000 is to buy a savage FTR in 6.5x284 or 308 ( I would go with the latter as you are not reloading yet I assume). I love my 308 and yes it can be used for 1000 yard shooting but it is not the best at it either by a long shot. The 6.5-284 is a much better choice. Many of the 1k shooters replace barrels once or twice a season depending on how much you shoot. Then you have $2000 left you better spend every dime of that $2k on a scope if not a little more.
Leupold Mk4 and Nightforce are the lower end of the long range optics. I am Dillinger in that Smidt and Bender are the best there is and when I looked a few weeks ago they are all $2000+ for the scope.
Your best bet is to look for F-class matches in your area and go watch them. Talk to the shooters and listen to what they say. Most of the time some will let you shoot their rigs if you pay them for ammo and or bring your own.
I had a guy offer to let me use his rig in 308 as long as I brought 200 rounds of Black Hills 175gr match ammo to the match.
F-class has one groups F-TR that is limited to 308 only. This is a good place to start is makes things a level playing field as every one is shooting a 308. F-Open is a bit different and where you get into some high dollar rigs costing $4000+ just for rifles your looking at 32"+ long barrels and wildcat cartridges.