Moving MySQL databases.

I am currently with a hosting company that provides MySQL databases.  I'm an
ASP/SQL guy who's transitioning some things over to PHP/MySQL on some of my
hobby sites.

My question is how easy is it to move those databases over to another host
should I need to?  Is there a file somewhere that I can download via FTP?
It appears I can only access MySQL databases tables via the shell account or
via PHP scripting on my pages, so I'm not clear on whether or not there's an
actual database file with my tables that I can actually take with me should
I ever need to move to another host.

Can anyone with MySQL experience clarify this point?

Thanks,
Mike

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