Madpuma, this is a great place to learn a ton of valuable airgun information! Welcome aboard!
You've got alot of choices and alot of great responses from people. I'm sure you're a bit confused now....
I would like to help remove the confusion-
You mention a price range of $500, give or take $100. You can get a fantastic air rifle for that price, and even less. Longislandhunter, aka Jeff, gave you some really good details and info that I completely agree with. He mentioned 2 RWS guns that are both excellent options for what you desire. Both guns have the power to dispatch larger critters, have a great reputation, and quality is way above average. RWS guns are very accurate as well.
He also proposed the precharged pneumatic option, where you could buy the Benjamin Discovery and the hand pump that goes with it. This is another great option if you'd like to get a fixed barrel, non-springer air rifle. The Discovery has a great reputation on here and is very accurate and powerful. I've read many posts of guys taking 50+yd shots and making 1 shot kills on larger critters with this gun. Also, like Jeff mentioned, you don't have the recoil with the Disco that you'd get from a springer. So, they tend to be easier to learn to shoot quicker. You can pick up the Disco and pump for anywhere between $315-400, which would leave you enough $ leftover to get a good scope for it. You can pick up a Centerpoint 16x40 with illuminated reticle for ~$65 and it will serve you really well.
The Discovery, out of the box, is slightly loud. Most guys purchase a TKO aftermarket muzzle brake from here
http://www.tko22.com/. I've heard this brings the gun down to near silent.
So, it all comes down to what you want. My recommendation would be to get a high quality, accurate shooter out of the box as your first air gun. Hehe (I said first, because it probably won't be the last). Both of the RWS guns you mentioned fit the bill as high quality and accurate, oh and powerful! The PCP option is there as well with the Discovery if you want to go that route. I have the Walther Force and its a good gun for the price. It's a powerful gun and decently accurate if you were lucky enough to get a good one. It's a cheaply made gun and nowhere near the same class as the RWS 350 or 48 though. Since your budget is what it is, I would definitely rule out the Walther and look at the others. If you go that route, you'll have no regrets and a very satisfying hunting gun that is capable of dispatching a wide range of vermin.
I hope my two cents helps! Let us know what you decide, and again, Welcome to GTA
