Is it better to plug the monitor into the motherboard or graphics card, both have DVI ports
What???If you have a dedicated graphics card, it should be far better than the one on your motherboard. That's reason #1. Reason #2 being that you shouldn't be able to see anything with that graphics card installed if you plugged into your motherboard's DVI port.
Since you are using a headset I would say DVI, if you were listening to sound out of your TV then HDMI. The cables are essentially the same, even though the connectors look quite different. DVI does not support sound, but HDMI does not support resolutions above 1920 well. <S> HACK