The differences between component and HDMI are that HDMI is a digital signal versus a analog signal you have with component. Also, with HDMI, both the video and audio are included on the same cable so if you mind about extra cables, it is pretty nice to have to only plug in one cable. People can debate all day about which is better. I cant see a difference between the two on my Direct TV box. Some people have problems with HDMI because of the HDCP(copy protection) that is on almost every HDTV and hi-def device you hook up to it. Some times it dosen't wanna sync up from the your hi-def device and your tv which means you dont get a picture. Other than that, the only thing is the ability for 1080p.
Most hi-def devices only put out 1080p over HDMI. Your 360 Premium will only do 1080i max res over component. You will either need to get the VGA cables for the 360 to get 1080p output or buy a newer 360(all the new ones now have the hdmi port,even the arcade).
If you wanna get be ready for the future, Id say spend the extra money on a native 1080p set. I just got a new HDTV about 4 months and went with a 720p native res because I only wanted a 40inch and their were so many good deals(and still are) for 720p native sets. But the only 1080p content out right now is Blu-Ray and HD-DVD movies. And I dont think the tv networks are gonna switch from (720p/1080i) any time soon. Also, anything really under 42inches, you are most likely not gonna be able to see a difference between the two resolutions.