Hi Tim from Ft.Wayne...
I don't know what your requirements in an air gun are but I will be happy to opine regardless...
You can limit yourself to a $200 gun and probably get something that will dispatch a chipmunk or squirrel at 10 to 15 yards with a fair amount of consistency.
I did exactly the same thing about a decade ago. I bought a Gamo Varmint Hunter in .177 and at the time, believed it to be an awesome air rifle. Looking back, I so wish I had known at least the little bit I know now about air guns. So it is with that thought, I advise you to save your 200.00 until it reaches maybe 600.00. Buy yourself a Benjamin Marauder or a quality spring gun (AA-T200 or HW97K) and put a cheap scope on it (for starters). Now you'll have an accurate, fairly quiet rifle that will put pellet on top of pellet within the confines of most suburban backyards.
Spring guns I am learning, require a modicum of skill from the shooter. But if you like to shoot, they lend themselves to lots of practice. If you go with a spring gun, don't make the common mistakes. They do NOT shoot like anything else that launches projectiles. There are more than a few individuals on this forum who will kindly guide you in that discipline.
I hesitate to hit the submit button below, knowing full well I am in the early stages of "The Dark Side" addiction to air guns and I am encouraging you, a novice to "just take a little puff." Once you shoot something that performs better than you ever could have imagined, you too will be addicted. In that addiction as with any other, lies a huge outlay of money and excuses to the spouse... I'll be 60 in a couple of months. I hope I live long enough to acquire and shoot the tons of air guns currently on my wish list. I regret not having taken my first puff 30 years ago! Good luck my friend!
Oh...By the way, that Gamo I told you about... After owning my first PCP, I gave that Gamo to a friend. I couldn't bring myself to shoot it again. The sadness is that the novice friend thought I did him a favor. ;-)
Cheers,
Danny