for a varmint gun on a $1200 budget i would spend
about $600 on a good bolt gun with a heavy barrel and $600 on the scope.
probably start with a Savage 10FP, plus however much scope i could afford with the leftover money. i wouldnt even consider a semi-auto - for that kindof money you will not find one accurate enough for varminting.
or check out the clearance room at
http://www.frontiertaxidermy.com/html/clearence_room.html
there are some .223s there like the remington SPS varmint, if savage isnt your thing. $589 is a good price for an SPS varmint in .223. my varmint rig is a remington 700p but you wont get that for that price.
the remington SPS is basically the exact same rifle as the (current production) 700P, but with a cheaper stock than the HS Precision alum bedding block fibreglass stock on the 700P, Sendero, etc.
i would probably still go with the Savage 10FP though in .223, with the best scope you can buy. the stock is the only weak point on this rifle - but its just as good as the one on the 700 SPS and there are a ton of aftermarket options for the Savage. you can upgrade it later on if you decide to, but the current stock it comes with is not that bad for varmint use - its very light, pillar bedded, and with a bipod you will not run into the side-side forend flex issues that are its main weakness. aftermarket stocks are widely available from something like the Choate or a laminate from Boyds or Richards Microfit.
most importantly its accurate and has a great trigger that you can adjust very easily down to a crisp 1.5lbs, and its easy/cheaper to rebarrel later on if you want to change calibers, etc. you could change to a .243 (an amazing varmint caliber, though double the ammo cost) or something if you get tired of the .223 with a new bolt face, barrel, GO guage and barrel wrench.
**edit
i notice people recommending rimfire.
if i were to go with a rimfire id go with a .17HMR, not a .22. probably a Ruger. cheaper ammo, almost as good performance as the .223. however you are going to be more limited in range.
i have a 10/22T with a VQ trigger and Bushnell Elite on it that i use as a 'loaner' when i take friends who dont have a varmint gun varminting with me (many just have something like a .30-06 700 CDL for deer, not exactly a varmint rig). youre not going to get much better than that in a .22 given the option. they are happy with it for varminting, but compared to the .223 700P its a joke in terms of killing power, accuracy, effective range. to sum it up i would
not want to be using the .22. not to mention its not always a guaranteed kill, even with stingers.
in Ontario i will bet my life you will not find a place to shoot groundhogs where
'melting your barrel' is an issue, especially not a heavy barreled rifle like these. i go to the most groundhog infested areas each year and even the farmers that LIVE THERE and beat me to the early season groundhogs dont kill that many. there simply are not that many groundhogs in Ontario for this to be an issue anymore.
15-20 years ago, sure id believe it, but not now.
here is a review of the remington 700 SPS, much of it comparing it to the Savage 10FP:
http://www.snipercentral.com/remspsv.htm