After much researching I bought a Gamo CFX in .22 and have no complaints. I have not hunted anything worth mentioning, however the air rifle is amazing. The CFX is an underlever which in theory should be more accurate than break barrels. So I would recommend perhaps considering an underlever unless a break barrel is what your set upon. The Truglo sights work very well, however for greater accuracy I placed a leapers scope on it and shoot 30 yards for target practice, with normal groupings of 1/2 inch or under for a series of shots. The velocity is rated at 750 fps which should be plenty when pushing a 14g pellet to take larger game. The price of the gun would fit your budget perfectly with a nice scope.I'd be happy to answer any questions you have as well.
Thus being said the Gamo Whisper in .177 should be adequate for most game with proper shot placement.