I have a class website for my students that I've created using Microsoft Publisher 2003. Whenever I work on the site I have to save it as a web page and then upload the file to my district's server. The site has always worked well on Internet Explorer (I should hope so, they are both Microsoft products). My site used to work on Firefox and Chrome, but the page rendering made the page look a little fuzzy. Now that I've upgraded to Office 2010, my website will render perfectly on Firefox and Chrome, but many of the links will not work.

Can you think of a reason why this might be happening?

Would you mind posting a link to your web site? I can look at it and see why it isn't working.

There are bunch of things that could be causing this. Naturally there could be many solutions. Though, one simple thing You could try is downloading the Open Office from http://download.openoffice.org/index.html , install it and try to open and export your file with this application (it's 100 % free and contains no spy software or viruses). It might work, it might not. But that's the most simple thing I could think of.
What I would recommend though, is getting more intimate with HTML and CSS as a start and then begin coding your website in simple html editors - doing it this way You ensure most compatibility between browsers and the different versions of them. That brings me to another point, that a broken web page could result from updating Your browser as well.
There is no easy way when we are talking about securing browser compatibility- this is a "science" for it self, which is why You can find so many websites online that focused on such issues.
I don't recommend using Microsoft Office for managing websites, may be with the exception when You are dealing with .asp technology which I am not proficient with.
Hope that helps.
PS: If You decide to follow my advice I can provide You with numerous links to websites with basic and advanced html and css tutorials (by the way You only really need html, but css makes things prettier )