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As far as I can tell, the transition of domains was pretty seamless since I had checked out a lot of what I had to do beforehand. There's things you have to be cautious of to copy over existing databases (of stories, comments, etc) and also to avoid losing search engine ranking.
Also a couple of things I forgot: If you were using a Google Maps API, you have to get a new key or your map won't work. If you're using MySQL, check the timezone and ensure it's the same or you correct for it. Check pages (and particularly any templates you use) to make sure your links were either relative or are updated to point to your new domain - links should be relative...that makes this step easy.
Read more for details on the domain hosting (and registration) transition process.
If you are looking to switch domains, you should use 301 redirects to replace old content with new. This way, search engines will know to point to the new information without penalizing you for duplicate content or broken pages. Try this site as a good starting point. (I just googled "301 redirect htaccess."
Copying a database in MySQL is pretty straightforward using the mysqldump command, which automatically generates the code to recreate the tables.
Transferring the actual domain registration is not complicated, but takes up to a week. The way to expedite the process is make sure your domain is unlocked and to change your DNS entries to whatever your new registrar specifies. Doing this, my transfer took 5 to 6 days.