300 Win Mag isn't going to be a cartridge you want to fire often enough to be proficient with, just a lot of recoil and blast for little gain, and the belted magnums,since they head-space off the belt instead of the shoulder "can" be less accurate in the long run. 308 based cartridges are normally given the edge in repeatable,consistent accuracy over long action cartridges like the belted mags and 30-06 because of the powder column/load densities with the given cartridge shapes. The 223,22-250,243,260,7mm-08,6.5 -284 and 308 fit your parameters
223 is more economical to shoot than most other centerfire cartridges.22-250 is a very fast cartridge and has a reputation for burning out barrels quickly,same with the 6.5-284. The 260 and 7mm-08 have some of the best bullet selections for Ballistic Coefficients and sectional densities-long/slender bullets as compared to a 308,where 168 to 180 grain is near max weight because of Cartridge Overall Length (COL) and powder load compaction.
The Remington 700 does have merit, but I would get a CZ 550 in 260 or 7mm-08 if I were shooting for target only; and like the Remington, you can get it in Varmint Specific trim, which is what you would want for target shooting. A Sako 85 would also be a very fine choice. The CZ also come in a small bore rifle, the 527 for 223,22-250,204. CZ is a world class rifle,as is the Sako.
I have Browning A-Bolts, which are fine for accuracy but aren't normally what people use for target specific shooting. I bring this up because I handload for my rifles and have gone from having near 2 inch 3 shot groups to 5 shots I can cover with a quarter (100 yard range) in 30-06. So, handloading will improve your groups faster than any other avenue.
My server isn't calling up the CZ home page right now, but it is worth the look. The CZ Page is being updated!!!!
EDIT: 260 and 7m-08 are both necked down 308, and again,308 with 165 or 168 grain Match grade bullets would be great too. At ranges less than 300 yards the Mil-Dot or Bullet Drop Compensation scopes won't even come into play, just not enough variance. Since I'm set up for hunting deer and elk more so than targets, I have a 200 yard ZERO and my 3x9 50mm scope is near perfect for that.
At $200, I like the Nikon Pro Staff 3x9 50mm with the Niko-plex reticle, and the Leupold VX-I 3x9 40mm for about $250 (retail each),but if you look on line you can find some better Monarch or VX-II on sale.
To take advantage of a Mil-Dot or BDC scope you are out past 300 yards for range--and you would really want to spend more on a scope. Anything over 300 yards and your optics need to be up to the task.