I have a Marauder in .22. Nice 75 yard gun. Shoots 18 grain JSB pellets about 830 fps for about 28 foot pound energy, which is enough for just about any critter from birds to rabbits. Accurate at 75 yards, nice 10 shot groups at about 1.5 inches. The only thing I did was adjust the transfer port so I got the speed I wanted. Other gun is an Impact, but way out of price range. The Marauder will do wonders on the critters.
Another option might be the Fortitude, but, it's tuned so the 18 grain will fly somewhere in the 700 fps to 800 fps range. Not sure what that'll do for accuracy. Check the reviews, excellent gun for $300 is what we are hearing. Marauder is a bit more expensive, but the trigger is worth the difference. Fortitude trigger = crap, Marauder is an excellent trigger, two stage, makes shooting enjoyable.
Money - On the Crosman Web Site, Marauder is $539, you will need a scope as well. Use the AGNATION discount code for a 25% discount which brings the price down to $405. Most Fridays are Free Shipping, saving another $20 or $30. Scopes and other Crosman goodies also get the 25% off treatment on the Crosman site.
You can pick up a hand pump (no fun, but gets you shooting) in eBay for $50 to $75. Recommend tanks, but that'll put you over the $600 budget.
People may tell you Marauder barrels have problems, but Crosman fixed that years ago. Mine is a tack driver. Very happy with it. Highly recommend the Marauder for an entry level gun, plus, extra magazines are $16 each. Buy 3 or 4 and shoot rather than reload.
If considering a Gauntlet, do some research. There's been some debate on the internals lately. However, refurbished Gauntlets can be had for $200 if you search for "Refurbished Umerex Gauntlet". Some people swear by them, some swear at them. I've considered one, but for now, I'll wait.
Keep us posted on what you buy.