Go with a .22 if you're going to use it on game. Ignore the velocity. Its foot pounds that matter. A .22 will ALWAYS hit harder than .177. The velocity there will give you penetration, but not the knockdown power. As for a $200 gun, I'd go to Airgun Depot. They are a great company. They also sell refurbs from the major brands. I'd look at RWS, Gammo, Benjamin and Remington in your price range.
As to pellets (and guns for that matter), screw local. For .22 you can order a 500 round tin of Crossman premire (good enough for government work) for nine bucks. You can spend a lot more on pellets if you want one hole groups at 25 yards, but I can hit a golf ball at fifty yards, and that's good enough for me. Though, air guns can be finicky about ammo. You should try a few different brands before blaming the gun if accuracy isn't what you expected. Check these folks out. And by that I don't just mean look at their web site. Do that, but then call them after you have some knowledge. They love to talk air guns. If you hit them at a slow time you'll have a sweaty ear when you hang up. Pyramid Air is good also. Buy with confidence from either one.
If it were me I'd look at either a nitro piston or a gammo silent series if noise was an issue. But the old school RWS models are built like tanks and scary accurate. Plus they are wood and steel and put together by anal retentive Germans. You grandkids will be shooting those rifles. I have an RWS 34 in .22 and it will bring an honest 800 fps to the table (for comparison a .22 short is about 900 fps) . To put it in scary terms...5/8th inch plywood is NOT an adequate back stop (don't ask). You want to hang some old rugs over a dowel or use a pellet trap for that. The only down side is that its a bit loud. A nitro Piston or a Gammo will be quieter, but they are plastic guns and I wonder about longevity. Anyway, talk to the folks at Air gun Depot and Pyramid and they'll steer you right.