Here are some great tools and links for ensuring that web pages work across a variety of browsers. The depth of testing of course depends on the type of traffic and scale of the project. In most cases testing for IE6, IE7, IE8, FF2, FF3, Chrome and Safari is going to be sufficient. Heavy traffic sites, .org or .gov domains and sites with specific usability concerns may need more!

Tools of the trade:

IETester, Multiple IE, W3C Validation

These are the main tools you're going to need. Once the page is built (using Notepad++ and Firebug of course!) and works in Firefox 3, we need to begin checking first on IE7/8 and then IE6. One day web designers the world over will rejoice when IE6 finally dies, but until then as many as 30% of visitors may still be using it and we have to support it. .net magazine has started a campaign to bring down IE6 once and for all! If enough high-profile sites like youtube drop support for it, it will hopefully fall out of use.

Although Safari and Chrome are generally quite close to Firefox in their rendering behaviour, we still need to test them, preferably on a mac and a pc! One thing to know about Safari when using matching image and Css background colours:
Because Safari is actually one of the few colour-managed browsers, image background colours and Css background colours may not match up. There is a more detailed description of the workaround here.

It's a good idea to make your site's Html and Css W3C valid. See my post on semantic Html for advice on best practices on how best to code using Html.

For watertight testing, some of the web-based screenshot sites are a very useful service. You can pick browsers and even specific OS to see how your pages will look. A great subscription service is Browsercam if you are doing a lot of testing or have a company account.

Useful links
http://freesitevalidator.com/
http://jigsaw.w3.org/css-validator/
http://validator.w3.org/
http://spoon.net/browsers/
http://try.electrumsolutions.com/demo/sortsite.aspx
http://freesitevalidator.com/