New gun

With that budget, you can't go wrong with either an Umarex Gauntlet or an Air Venturi Avenger. Be aware that both of these are PCP (pre-charged pneumatic) and require a way to put air into the gun. If you're not already prepared for this, it can be a large factor in your decision. If you are NOT ready to get into PCP guns, then I'd look at break barrel rifles. These are self-contained and all you need is the pellets to get started. Reputable manufacturers are many - from Weirauch, Feinwerkbau at the top to Walther and Air Arms to more affordable solution like Beeman, Diana, and Benjamin.
Be forewarned, though. Once you start... you can't stop! This hobby is HIGHLY addictive.

You can go CO2 with the chinese made QB78 and all the rebranded ones.
No need for pump, good power and accuracy.

Do not expect bench rest results with cheap guns/sights/ammo bu they are tools to do a job, and they do it well.

CO2 is alright for plinking or teaching someone how to use a rifle. But, I don't recommend CO2 rifles to anyone that wants to hunt or do any serious shooting for three reasons: 1) lack of power - the Beeman QB78 is advertised at 650fps and we all know that manufacturers inflate those numbers. 2) inconsistency - every press of the trigger reduces the power available for the next shot. 3) Temperature fluctuation - cartridges are very susceptible to atmospheric changes.
Every piston powered rifle I have shoots harder than 650 fps. A couple of them can't even be shot in the back yard because they break the sound barrier and the neighbors think I firing off .22 long rifles!
Just my 2 cents,
Mondo

Mondo is right about co2,You can't go wrong with PCP's, or break barrel, only problem in my view with break barrel is 2 fold, 1st: most are very finicky about how you hold them, and 2nd: they destroy scopes. you will need to spend darn near $300 for a scope that can handle the punishment dished out by the spring systems i picked up a Beeman Chief2 Plus-S(3000psi) for 179.00 shoots great, stock setting moves crossman premier 14.3 @ 830fps and you can turn it up or downfrom there, and it is pretty darn quiet to boot. I have it, I highly recommend it for an entry level .22. Decent, inexpensive hand pumps can be found for as little as $50 or $60 (just don't over heat it!). i have seen some great reviews on the ebay pumps.
good luck and happy hunting