You say "sniper rifle" which is usually a military term to shoot enemy soldiers from concealment at a distance.
You want a bolt action rifle in .308 Winchester caliber. I suggest a Remington 700 with a 20 or 22 inch barrel with a 1:10 or 1:11.25 inch twist. I prefer HS Precision stocks. The rifle may have a Remington XMark Pro trigger, which some people do not like; I am on the fence about them. I would stick with an internal magazine instead of a more expensive box magazine. It would be nice if the barrel was already threaded for a suppressor (NOT a muzzle brake) which you may consider in the future. I would put on a Harris or Caldwell bipod. You will need a picatinny rail for mounting the scope, and either 1" or 30mm scope rings. I like Warne scope rings, or Badger. I prefer Nikon or Leupold scopes, or a SWFA Super Sniper, with mil dots or something like a TMR reticle (Leupold's tactical milling reticle, also comes illuminated). I believe a fixed 6x or 10x (at the most) scope is all that is needed for military sniping; you are looking at hitting an 8-10" circle at 600 yards or so, not shooting 2" groups. That is what dedicated target/bench rifles are for. A smaller magnification gives you a much better field of view, which is what you need as a sniper, to find, identify and track targets. You should be able to shoot this rifle standing, kneeling, prone, barricade, off the bipod, and off improvised rests -- that is why I object to magazine fed sniper rifles. Magazines are good for competitions; plus I like to be able to adjust my hand placement depending on my position.
I have a Remington 700 LTR in .308 that I purchased used for about $600. It has a variable 3.5-14x mil dot Nikon ProStaff 5 (about $400) on it right now because I am using it mostly for load development and appreciate the higher magnification when I am shooting groups that may be sorted by 1/10th of an inch at 100 yards before I settle on that load. I can leave it set at 6x; 10x is the power for ranging with the mil dot on that scope. It has Warne rings and a Warne base (about $100 for both). The factory trigger (not X Mark) breaks consistently at 2.25 lbs. on my digital scale. The factory barrel is a 1:10 twist, fluted, and is free floated. With the scope and sling it weights 10 lbs. unloaded. It is a joy to carry and shoot offhand or from improvised rests. With my handloads I have put 10 rounds into an inch at 100 yards.
.308 allows you to practice offhand shooting with military surplus ammunition. The recoil is nothing and you won't burn your barrel out like you do with a 300WM or 338 Lapua. Practice makes you better, not fancy equipment. Learn to range distances through the scope and dope the wind. Chrono your loads if you can and run them through a ballistic calculator so you know your drops at distance, and tape that to your stock. Or you can chrono your loads and have Leupod or Nikon make you up custom scope turret caps($60) when you can dial the yardage and hold on the crosshairs, no dialing mils needed.
I am really a keep it simple stupid kind of guy. I have spray camo on mine; I used local leaves and grass as a template. I like to try and shoot every week. Save your money on a custom gun and buy a similar 22LR bolt gun as a trainer; it will help you read the wind when shooting at 25 and 100 yards.
Bottom line: Remington 700 LTR, or a 5R, in .308 Winchester with a 20-22" barrel, Warne mount and rings, Nikon, Leupold or SWFA scope with a simple reticle, Harris or Caldwell bipod, simple sling, buy 500-1,000 rounds of ammo and practice in all weather, year-round.
Edit to add: before I had the variable Nikon mounted I had a fixed 4x and could hit whatever I wanted. I will likely go back to a fixed 6x, maybe 10.