I'm going to assume that what you have for a TV is a 1080p, 120 Hz, LCD, LED back-lit display. If your DVD player does upscaling to 1080p, it would be via the HDMI output. You then need to select that you want 1080p HDMI output on the DVD player.

If the DVD player does not upscale to 1080p, you may be better off using the 480i or 480p (trial and error as to which is best) output via the component video output on the DVD player. Again, you'll need to select 480i/p. The TV will then upscale that signal to it's native 1080p display.

Don't forget that some dics look better that others, so try a few.

BTW - High speed HMDI vs Std HDMI is a somewhat questionable rating given to the cables when length exceeds 25 feet.

The short answer is if you want to take full advantage of your new TV, then you need a Blu-Ray player. DVD's have far less information on them than your TV can display. To make this simple, lets assume that your TV has a million little pixels. Your DVD only has information to fill, say, 700,000 of those pixels. In a process called "up convert" the DVD player, TV, or other device basically fills in the missing pixels, kind of guessing what should go there. Some devices do a better job than others but in the end a DVD will never look as good as true HD.