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How Do I Know What To Buy?
Hope this helps to answer some of your questions. As far as rifles are concerned; you will need a caliber that provide bullets with good design and ability to shoot at long distance. A scope that will provide optimal clarity with enough adjustment to allow you the distances to reach out there.
With that said, here are some thoughts. With the budget you are looking at spending there a few rifles that will work. I prefer Savage rifles, but Remington and Howa offer great starter rifles to fit your budget. Weatherby offers the Vanguard $399 that is mounted in a Howa action. Between the four, Savage and the Howa would be my top pics. If you can get into a Savage with an Accu-trigger with pillar bedding you will be ahead of the game. Howa's with a trigger adjustment by a smith for around $20 or $30 will do wonders and shoot very accurately.
Calibers to look for, for starting out. We know the budget, so let's consider application. If you are planning on reloading you are opening up a world of options. If you are shooting factory loads then we need to consider what is available.
If you are shooting factory ammo, then you want to consider the .308 (Black Hills, Federal Gold Match, Hornady Tap ammo, HSM), 243 Win, 260 Rem, 6.5x55 Swede (available in the Howa, in my opinion a very good long range starting caliber), 7mm-08, 280 Rem, 7mm Win Mag, 300 WSM, 7mm WSM, 300 Win Mag. These calibers all have very good factory and production loads that will allow for good medium range ballistics for hunting deer size game and 1000yrd target (.308, 6.5x55 the exception with loads capable to 800yrds and all magnums capable to 1000yrds for harvesting game). I would not suggest the any of the Magnums unless you have two things; experience shooting them, or a muzzle break to tame recoil. Recoil can be enough to scare any new shooter and start bad habits that will frustrate and cause in accuracy.
If you are reloading or planning to reload, then you can look at others like the 280 AI, 6.5x284 Norma, a tight twist 22-250, 223, (both low recoiling, but with tight twists offer very good heavy bullets for long range varmint and target applications). The 243 in a 9.5 twist can handle the 105 Hornady A-max and 107 SMK with ease and good ballistics down range. Moving into the 7mm-08 and 260, 6.5x55 there are phenomenal bullets for long range application. Then you get into the .284's/7mm and .308 calibers that are endless in your ability to chose a bullet and shoot.
Scopes are a beast in them selves. Everybody has a favorite, but for the price range you have specified let me make these suggestions. I own 2 Buckmasters series Nikons. For under $300 dollars, they are in my opinion one of the best pieces of glass that can be purchased for long range hunting and target. I have found the 6-18x40 a great scope and capable of seeing ground hogs at 1100yrds with ease. Another that is getting great reviews is the Vortex Crossfire. It has plenty of elevation for any distance you want to shoot with 120MOA listed in their specs. With the proper set up you can get 40 to 50 MOA of adjustment with the Buckmasters which should be enough using low mount rings and 20MOA single piece (tapered for more scope adjustment) base.
A lot of guys are liking the Bushnell Elite series. I have used the 4200 Elite Tactical, but it will be out of your price range with including the cost of the rifle. The other option is the 5-15x40 3200 Elite. They run these scopes on the Barrett 50 calibers, so that should tell you the robust construction of that scope.
So that should get you started. Just ask away and we will guide you as you go.