Check the Easy Stuff First!
While the difference you see between browsers might be due to some obscure browser bug or some new technique you're using, nine times out of ten, it's just something simple. It's easy to blame a problem on a new technique and spend hours debugging it, only to find that you're changing one file but uploading and viewing a different one.
The general easy stuff:
On the following three pages, I'll give you some ideas of how to check the easy stuff in HTML, XHTML, and CSS.