The web is a visual medium. Well, mostly.
There's no better way to test a visual medium than by looking at it. Look at your site in as many browsers as you can. If you've already got as many browsers installed on your development computer as you can fit, get another computer and install some more. Either that or run a Virtual Machine.
If you can't do that easily, you could use one of the growing number of browser testing services. These are server rooms packed with computers running Virtual Machines and automated systems to which you supply a URL, wait a few moments and get shown an image (or several hundred images) showing your URL in different browsers on different platforms. Some of the more sophisticated services allow you to scroll down a long page or specify different actions, text entry or mouse events you want to see triggered. These services can be exceptionally useful when it comes to developing HTML e-mails as there are some rare and esoteric e-mail clients out in the wild. Litmus does an excellent job at virtualised testing for HTML e-mails. On that note, the Campaign Monitor library of free HTML e-mail templates is a great place to start, learn and possibly finish when working on an HTML e-mail.
There is also a place for automated testing for some things. Recently, there has been a bit of a movement away from validating code as the purpose of web development is not to make it 'check a box' on a merely technical level, it is to get the message across via the design however possible. However, validation is still the best and easiest way to check your syntax. Syntax errors are still the main cause for mistakes appearing in your web sites and are the easiest thing to fix. Don't assume IE is wrong. Again, if you're keen on HTML e-mails, here's a great post on the Litmus blog.