Another good idea is to gain a working knowledge of the operating system for the server on which your Web server is running. This facilitates easier monitoring and tuning for performance problems. If you are a z/OS DB2 DBA, this will likely require you to get out of your comfort zone and dig into Linux or some variant of Unix.
It can also be worthwhile to develop a better understanding of highly available RAID storage technologies in use for most 24x7 Internet applications.
Of course, having a clear understanding of the business functionality that is being achieved via the web-enabled databases in your organization is also vitally important. Only by understanding the business impact of database downtime can the appropriate administrative techniques be deployed to maintain constant availability.
I don't imagine that this is a 100 percent complete list of thing-you-need-to-know, but it is probably a good place to start your learning adventure. Let me know what I might have missed by adding your comments below!
And if you already support Internet access to your databases and you don't feel up-to-date on these items, then it is definitely time to get cracking!