You should, of course, perform the basics 1) a Web page test, 2) HTTP headers test and 3) link check. You definitely want to check the rendering of the site across various browsers and operating systems. There are some handy tools online to do this such as Browsershots.
Websites are basically client/server applications - with Web servers and "browser" clients. There are applications that run in Web pages, those that run on the server side and so on. With seemingly endless "moving" parts and connections, what are all the various ways that Websites should/could be tested?