I'm looking for my first break barrel pellet rifle for hunting
 

For the purpose, I think you need a higher caliber (.22)

I was thinking about a .22 my friends and I hunt with 2 .177 gamo's and the rabbits get away. Do you have any recommendations on a strong .22? I don't really have a budge but I am not looking to get a $500 dollar gun. Only about $300. Also, What ammo do you recommend?

Thanks for your help.

Seems like, .177 will do just fine according to these articles.

American Airguns - Field Use of an Airgun
Compared

...especially if you use heavier pellet.

I like Beeman Crow Magnum Amazon.com: Beeman Crow-Magnum .177 Cal, 8.80 Grains, Hollowpoint, 500ct: Sports & Outdoors

Check these .22s

Walther Falcon Hunter Edition- .22 cal 1000 FPS Air Rifle (3-9X44 Scope)

Benjamin Trail NP XL. Air rifles - PyramydAir.com

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.