Welcome to the forums.
I think you do not want to use the speakers from the monitor so HDMI or DVI eider one will do.
The quality of the image is about the same.
If you want to use the HDMI port you need a HDMI cable, if not, just use the DVI cable that came with the monitor.
I have 2 monitors, one connected with DVI and the other one witch is LCD TV with HDMI cable.
The HDMI cable is not that expensive if you want to buy one
"I was wondering if I need to buy a separate HDMI cable to connect my monitor to my graphics card or if I use the DVI cable the monitor comes with will I be able to get HD quality from that? " -short answer, yes!
EDIT: HD refers to video having resolution substantially higher than traditional displays. The video card can support a much higher resolution than 1080p(1920x1080) with a digital signal(meaning HDMI or DVI).