Migrating emails from GoDaddy to new host

Hey, not sure if this is the right place to ask this. Forgive me if I'm posting in the wrong place.

I am switching from GoDaddy to a new host (haven't decided which yet). I need to move the emails from my GoDaddy account to the new hosting provider.

What is the easiest way to do this?

I appreciate all suggestions.
Thanks,

What mail are you using? Google apps, Roundcube, etc?