Wireless is a myth. Speakers need power.
At your budget, you are in the HTIB (home theater in-a-box) range. So, something like the Onkyo HTS-780 system may be about all you are looking to get. There are a couple of systems with wireless rear speakers... Wireless meaning you still have to plug them in at the rear of the room. But, I know very little about them.
Any A/V receiver allows you to plug your gaming into it. Video can be run directly to your TV - and I'm sure you aren't looking for a TV as well at this price point. So, the X-Box video goes to the TV, and the X-Box audio goes to the audio system.