Forgive me if I'm not the first to say it, but what you really want is a .22 LR bolt-action. You can get a good one for a couple hundred dollars and it will shoot better than you with iron sights. You'll never outgrow it. Even "match grade" ammo goes for less than $0.10 per round.
Start at 25 yards. Learn the fundamentals of marksmanship. Before you're through your first brick you'll be putting 10 rounds in one ragged hole. Any stray shots will be your fault.
Ultimately, it's the most useful rifle you will ever own. With a .22 and a brick of 500 rounds in a shack in the woods, you'll be fed for a year.
If you ever go way off the deep end with the libertarian survivalist "lifestyle" you can even manufacture a simple, quiet silencer complete with three hots and a cot in a federal penitentiary that won't affect accuracy with basic hand tools. Silencertalk forums and some Finnish sites have plenty of info on this.
Some quality options:
Savage MkII with synthetic stock $197
with wood stock $197
Cz 513 BASIC $290
I like the Mosin Nagant and own one myself. It's dirt cheap and will outlast you. There is no breaking one, ever. Also, some of history's most killingest snipers used them in sub-zero conditions with iron sights to kill hundreds of commies and fascists at insane ranges.
That said, you can't go wrong with a modern scoped bolt-action. The .30-06 is a fine caliber that was designed to fly most of a mile across a nice, flat no-mans land to the opposing trenches and punch through several inches of thick winter clothing and kill in a single shot. It's also basically the same as the 7.62x54R the Mosin fires. While it's a lot more expensive, you don't have to buy it in cans that were cached in cosmoline for decades in preparation for Nuclear Winter. High quality, precise ammo with payloads from super-match grade to frangible or expanding hunting bullets are availible in every gun store and Wal-Mart in America.
First, a rifles accuracy is measured in Minutes Of Angle (MOA). It's about 1" at a 100 yards and 2" at 200 and so on. If you're rifle shoots 1 MOA it means it groups (3, 5, 10 or however many consecutive shots) inside a 1" circle at 100 yards. Most any good, modern scoped hunting rifle will do this if you can. Put in your time with the .22, learn basic ballistics (I suggest you torrent a copy of Maj. John Plaster's Ultimate Sniper), feed your rifle quality ammo, clean it often, and you'll be killing things at 500 yards and further in no time.
Basically, all standard bolt-actions you'd consider are mechanically the same. The technology has changed little since the Mauser came along over 100 years ago. Several companies make really accurate, rugged rifles of this type for under $500 before scope. A great scope can be had for $200-300, sometimes less.
For rock-bottom price and excellent performance, you probably want a Weatherby Vanguard or Stevens 200. They've got good guts but no frills of any kind. For a couple hundred more you should look at the Remington 700 or Savage with Accutrigger.
Weatherby Vanguard .30-06 $382
Stevens 200 .30-06 $322
For scopes, check out the Burris FullField II, Leupold VX-I, Nikon Prostaff, and SWFA Super Sniper.
Burris FullField II $320
Nikon Prostaff $159
Leupold VX-I $320
SWFA Super Sniper $320
The Super Sniper is basically identical to the Unertl scopes American snipers have used since Vietnam. It's only real drawback is that it is fixed power, so the magnification is a bit much for the closer shots a hunter often makes.
Happy shopping and happy hunting.