Even though both of my wheels are tightened all the way, the wheel itself is loose.
I can slightly move it back and forth from left to right because there is a bit of play.
What can i do to stop this, or is it the wheel itself that's bad?

Tightening the axle nuts has nothing to do with your problem. You need to adjust your cones.

http://www.parktool.com/repair/readhowto.asp?id=105

On the inside of the dropouts there will be nuts that you have to tighten. Don't overtighten them because then you're wheel won't spin.