HDMI is technically superior to component but the difference in picture quality is negligible. Personally, I'd save the money and stick with component but it's up to you.
Just to be sure that you have it configured correctly:
The cable that came with your 360 should have red, blue and green connectors (video) and the red/white (audio), disregard the yellow cable. Hook them up to the corresponding jacks on the TV labed "Component" - make you set the switch on the cable to HDTV. In the 360's display settings, set the display mode to 720p.
I'm guessing that you have a 720p LCD TV, since they're the most common, and if you had a 1080p tv you'd be bragging about it
