I'm trying to migrate my wordpress site from an old host to a new host. The Wordpress Codex says "you can move by just copying your files and database." Copying the files is easy enough, but how do I copy my database? I'm sure I'm just not looking in the right places, but I can't find any documentation on this anywhere.
The same WordPress Codex has a page for backing up your database with phpmyadmin; see http://codex.wordpress.org/Backing_Up_Your_Database
You must export and import the database itself - and not do a posts export from the dashboard - to retain all settings, image URLs, etc.
Either use phpmyadmin in your host control panel, or you can use it as a plugin called Portable phpmyadmin: http://wordpress.org/extend/plugins/portable-phpmyadmin/
But if you use the plugin, you may have to try an older version, as the newer alpha version doesn't work on some web hosts due to host restrictions.
And, if you are changing domains as well as hosts, you need to correctly change URLs in the database. interconnectit.com WordPress Serialized PHP Search Replace Tool
Most hosting companies have a mysql viewer like phpmyadmin installed which lets you export the database. You can also export it by going to export under tools in the admin panel although this will not export some settings (plugin settings), so it is much better to use phpmyadmin if you can.