Sadly, it is not as simple as tag compatibility, none of the browsers implement the DOM properly and most render in slightly different ways, so there is no single solution. The thing to do is to decide which browsers you want to support: IE, Firefox Chrome and Safari are the obvious choices, but you also need to work out which versions of these you want to support. Then you need to try your page out on each, historically, IE was the quirkiest (still is IMO, but much better than it was), so a website written for that won't necessarily render in other browsers well. The rule of thumb is that the simpler you keep the HTML and as long as you style with CSS you have a better chance of success, your rendered HTML looks pretty good, though things like widths are better defined in css.
I skimmed
this[
^] and it seems to be a good guide as to where to start.
Your site renders OK in IE9 by the way, have you used HTML5 or CSS3 by any chance?
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