MS Component/Composite Cables have a SD/HD switch on them. It needs to be on HD, you also need to go into the Xbox 360 settings and change it to a HD setting. Not many HDTV's accept 1080P over Component, only HDMI, so you may need to try setting the Xbox to 1080i. With the 360 now showing on your HDTV, Hit the Display button on the HDTV's remote, or whatever it's called for the HDTV to show what Resolution it's currently in to make sure it's in a HD Resolution, and not SD.
Personally, when I got my newish Plasma a couple years ago, I ended up just selling my old Xbox, and 30 minutes later having gotten my new Xbox with a HDMI port. Much better, and simpler. Cuts down on cable clutter also.