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.
My husband and I recently bought a new Samsung LN46B650 and a Sony DVP-NS710H DVD Player. We have started to watch some dvds we own and have had pretty good luck until we watched Harry Potter 6. The picture is awful. It is extremely grainy and the objects/people in the background are blurry. I tried some settings on the dvd player and nothing worked. Not sure if its the player or the tv. Please help.