I recently moved my personal website from GoDaddy to my home server. I have a business connection at my house, and my site gets little enough traffic that hosting at home on my static IP makes sense. Moving the files wasn't really difficult, I FTP'd them down from the old server, and SFTP'd them up to the new server. Moving the database was a bit more challenging, however.
If you have shell access, it's a pretty simple process. On the old server, type:
mysqldump -u username -p databasename > databasebackup.sql
You'll be asked for the password assigned to "username", and then mysqldump will create a file that contains all the information needed to restore your database. One thing to note, however, is that going between different versions of mysql can be problematic. That's where the --compatible flag is handy. You can specify what type of database software you'll be importing to, and mysqldump will (try) to give you a compatible file. Some options are mysql323, postgresql, mysql40, etc. Check the man page for more options and explanations about what they all do.
To restore your database file on the new server, it's just as easy. Simply type:
mysql -u username -p newdatabasename < databasebackup.sql
That should transfer your data simply and easily. If you get errors, you might have to check that --compatible flag, or even do some more work to your database in order to make it compatible. One of the frustrating things with GoDaddy, however, is that you don't get shell access to your hosting account. Since my account was disabled, any mysql tools that might be available via their website were also unavailable. That's why it's important to have some backup software running regularly on your website. I was able to take an automated backup from a week ago, and simply import it into my new server.
The moral of the story, like most, is that backups are VERY important! It's great to know the tools to make a dump of your mysql database, but if something is corrupt, you'll want a backup rather than a fresh dump. If you have any other tips for moving databases from one server to another, feel free to leave them in the comments.