I can only give you some information but not a monitor/TV recommendation if you live outside Germany, because I am not familar with prices outside Germany :
You can connect all kind of monitors with your graphic cards, you can even connect a TV with HDMI connector to it.
If you already have an LCD TV or Plasma TV at home, you can use it as monitor without buying a monitor.
XBOX360 can also be connected to a LCD monitor if you have the right cable (you can buy this), which match your monitor, either HDMI or HD-VGA-AV cable.
A monitor or TV should do just fine for games.
If you are living in germany and really want a monitor : try to take a look at those monitors from Hanns G. They produces good quality monitors with fair prices.