Up to $500 includes a very wide range of guns. How far will you be shooting? Any chance of inadvertantly hitting another animal or poking holes in unwanted places if a pellet goes in an unintended direction for some distance? Is noise an issue?
For close range work the Beeman R-7 is good, as is the Webley Stingray/Xocet carbine. Both are top notch rifles with excellent accuracy, with power in the 12 ft/lb range. For longer ranges and more power, take a look at the HW77, R-9 and R-1. Those 3 are about the most popular adult air rifles going, and have power and accuracy to spare. On the quiet side are the Air Arms guns with shrouded barrels. The TX200 is a great rifle, but is real close to the $500 mark. There are many others, take a look at some of the dealer sites for prices and availability.
Theres no real advantage to the side-lever or under lever system as far as accuracy goes. Any of the better break barrel guns mentioned can shoot just as well as the side levers. Mostly these guns are far more accurate than the typical shooter is, so whether one gun is better than the other is really dependent on whos doing the shooting, not the gun.
Some of the side lever and under lever guns can be dangerous also. The ones that are loaded by sticking your fingers inside the breech to push a pellet into the barrel have been known to remove said fingers when the sear or trigger trips accidently and the breech closes at the wrong moment.
For optics, you won't do better than the Leupold 3-9x33EFR AO scope. At about $270 it's about the best value out there. Make sure you get good mounts too. Stay away from the B-Square adjustables and the RWS C-mounts. They are far to fiddly and don't hold up well on high power spring guns. Get something like the Beeman 5039 one-piece adjustable mount and you won't have any problems. Its a simple straight-forward no fiddling type thats easy to set up and won't ever move once its set.
Where in CT are you? I'm in Cheshire.
