Question:

What gives the best quality on my LCD TV (Full HD). I can input both component and VGA cables (and DVI, which unfortunately no console supports afaik).

In theory VGA should give the better results, because it's true RGB, which component is not. But I've heard some complaints about using the VGA cable on the 360.

Does anyone know what's best?

I would prefer to use a VGA cable so I can keep my Wii on the component input. I can always use a splitter of course.

I use the VGA cable and the quality is perfect on my CRT monitor.
Never had any troubles with it

Doesn't your TV have HDMI? That's what I use (for the same reason, didn't want to switch cables with the Wii all the time).

Buy the official VGA cable. 3rd parties not good.

Your display settings will give you a new set of options.

Try not to select a resolution that your TV cannot display because you might accidentally accept it, saving settings and you'll be screwed.  There is a way to reset it back to factory defaults but I had problems timing the button presses.

Also on the 360 display settings set the REFERENCE level to EXPANDED and you will notice the black levels much better!  The default is STANDARD and that looks very washed out.

Overall you will definitely notice that it is much brighter than component

DVI and HDMI are cable compatible.  Just get a cable with a DVI plug on one end and HDMI plug on the other, and then output sound seperately.  In terms of quality VGA and component are equal as they are both analog connections that split the video signal into three (RGB in the case of VGA and YPbPr in the case of component).  That's why the video quality of them is about the same and better than composite (which carries the entire video signal over a single cable).  HDMI and DVI-D are both digital, and therefore better theoretically than either VGA or component.

 P.S.: Don't pay big money on the DVI/HDMI cable.  A cheap one will work just as well as the rip offs.  (i.e., Monster Cable)

There's also DVI to HDMI adapters (and vice-versa). I'm not sure how you'd go about outputting the sound separately though.

This is what I would do. You can find a cheap DVI to HDMI cable from www.newegg.com. Make sure you've got the proper male/female connector on DVI end. Then you'd need to jerry-rig the component cable that comes with the x360 (remove the plastic housing) so you can have both the hdmi plugged in and the component plugged in at the same time. The component will be used for audio.

Sound output through the standard RCA stereo output (if sent to the TV speakers) or through the optical output (if sent to an audio receiver) works fine on every device I've tried.  I don't have an XBox360, so I'm not sure if there are any settings you will need to change in the system menu to tell it to output sound through the other ports instead of over the HDMI cable, but all of the DVD players and cable boxes I've played with do this without any fuss.  SInce he already hooked up his XBox 360 over VGA, he is getting the sound out through one of the other ports already.