You want to know how web servers work, how to configure them, how to check logs. You want to know about CGI, FastCGI, monitoring and maintaining servers. I would learn LAMP in and out.
Cross-browser compatibility is a big must. This is probably the hardest part of web development, learning browser bugs.
CSS - Do not skip learning table-less design.
Standards - Read specs on W3 on HTML, XHTML, CSS, etc Understand why standards are important.
Server side languages - Learn at least a couple of languages that operate on the server: PHP, Python, Ruby, C#, and Frameworks for these languages such as Django, Rails, ASP.Net, and ASP.Net MVC.
JavaScript - I'd get to know this pretty well. Very important for dynamic sites. You also want to learn JavaScript libraries like jQuery, Dojo, and Prototype.
Persistent Storage - Learn SQL, MySQL, or SQLLite, or MSSQL if going the Windows route. Maybe check out Berkeley DB if you're feeling adventurous.
Flash, Flex, Silverlight - At least stay current, maybe make something in them to understand how they work, security issues.
RFCs - At least know which are relevant. Read about HTTP protocols and headers, mime-types, etc.
Security issues - XSS, SQL Injection, form validation (never trust the user!).
User interface basics - Know what works what doesn't.
Source control - Learn SVN, and maybe git or mercurial
Design Software - Learn how to use Adobe products such as Photoshop and Fireworks as you'll be expected to make mocks into working websites if you ever work with a designer.