As far as I'm aware, there is no hdmi convertors for 360, only if they decide to release a Ver 2 box with hdmi which is rumoured (or BS) - see other link in this forum.
You can get cheaper hdmi switching but as krystalbouncy says, you could connect it using the current HDTV leads supplied from MS or Monster that are component and you get just as good picture imo and won't be missing out on much if anything on the picture quality compared to hdmi, unless of course you're going 50" plus HDTV!
I think you're confusing yourself with the terms. HDMI is just the type of cable that carries sound and picture. HD is the spec of the screen which should be a minimum of 1024 x 768 (720P) for High def. Wouldn't advise getting a 1080P screen as yet unless you;re getting it for a good price as you will only notice a difference if you're going for a really big size (50" +) and sitting a good distance away.