Help request: transferring a site

[amethysta]

I've volunteered to be the web master of our local beekeepers group. (Yay!) They are currently using Go-Daddy (probably Fat Cow for their greeness) and I would like to transfer them to a different host site when their renewal is up next month. Has anyone ever transferred a web site? Do I need to do it on the day of the expiration? Any help would be appreciated.