Morgan, to move the files it is as simple as using filezilla download files.
Then upload to the new site. You then will need to set the file folder permissions per the ubb install docs and edit the config.inc.php file again for the other paths.
Similar approach for the other web pages.
Then edit the dns record for your domain name so britbike goes to the new site. Or edit a redirect at the old site to goto the new site.
I would upgrade only after the move.
As far as the banner ,your issue is probably that it is using a domain name path and not a ip so until your domain name has migrated to the new server it will not work. Unless you know how to modify the config files for the banner.
