Today has been a flurry of sysadminlike activity, getting my new server up and running just like it should be. I did have to pester a number of people with annoying questions, mainly in the areas of DNS and email, but in the end I got one of my domain names transferred to the new server, a friend of mine to act as a secondary DNS provider, and email more or less working. I also got Apache, PHP, and MySQL working, and figured out how to set up virtual hosts in Apache. Things are running very much ahead of schedule, so hopefully everything will be moved over to the new server tomorrow or the next day.
Overall, everything has been easier than I would have anticipated. I spent quite a bit of time trying to figure out how to get email working, including quite a bit of wasted time on research, looking at solutions that are way too big and complicated for my very meager needs. In the end, a friend mentioned the Dovecot IMAP server, which turns out to be perfect for me. Both Courier and Cyrus are overkill for somebody who needs two mailboxes. I also spent some time reading about Postfix and how to configure it, but it turns out that just reading the comments in the Postfix configuration file was the best approach.
Running your own DNS server certainly makes you feel like a real citizen of the Internet. Everybody should do it once.