I've been working on updating the site to use Drupal 7. You might notice that at present, it's still using Drupal 6. There's a reason for that: it hasn't been exactly a smooth process.
First off, you're going to want to make sure you disable any custom modules and custom themes. They won't work initially. Then set your current theme to Garland.
So far, I've run into two empty tables that block the update process. I'm not sure what created them, but it's safe to just drop them: role_permission and date_format_type. (Update: Drupal 7 created them, and the MySQL script I was running to "reset" the database wasn't clearing everything.)
The SQL for that:
DROP TABLE role_permission; DROP TABLE date_format_type;
Once those are gone, I can run the update successfully. You'll get an error message about the timezone ID being invalid: you should ignore this. It's supposed to happen when updating from Drupal 6 to 7 and will be fixed by the updater.
Next up is to transition the theme and the custom modules I do have over to Drupal 7.