I'll break it down the best i know how:
Since cost is the biggest issue 223 rem is the right choice. Heavy .224 projectiles can reach out to 1000 yards but for this you need a faster twist rate (around 1 in 7 i believe) whereas most bolt guns have a 1-in-12 which lets you use up to 55grain projectiles. That's fine don't worry about that since it takes a good while to get out to 800 yards and finding a range that allows you to is another thing too.
I'd second vote that savage. Pac-nor and several other companies make pre-fit barrels for the savage for $300 or so which gives you the ability to modify down the line. Savage also has good aftermarket support with stocks etc.
What about an AR? - they're plenty accurate and also have a faster twist rate
Are you planning to get into reloading? If you reload the difference in price between major calibres is marginal. Even so id support the 223 because you can shoot it all day long and it never gets weary. Still plenty of recoil to be fun though.