Either HDMI or component (red green and blue) is what you want. That will rule out hardware issues. Also make sure your DVD player is set to output at the best possible resolution (1080p if its an upscaling model).

My gut tells me that you are just noticing the limitations of DVD now that you have a larger higher definition TV. The Harry Potter Blu-rays looks astounding but the later ones can be quite dark so I can imagine that the DVD's for the last two might seem a bit grainy on DVD. The Blu-ray versions have much better contrast and picture saturation than the DVD's... which I also own up through Goblet of Fire.