Hey Paul!
I just use the WordPress Export feature under ‘Tools’ and export ‘Events’, then ‘Venues’, then ‘Organizers’ all separately, then in the new website use the import function to pull them in?
Yes, in general this process should work.
You mention not wanting to screw anything up, which I totally relate to, so just one last note here: despite the likely success of using the WordPress import and export features, you should make a backup of your site regardless. There are many ways to make backups and it’s just generally a good safety measure to do before site migrations, or before installing or updating any plugin or theme on your site, etc.
I hope this information helps – let me know if there’s anything more I can help with! 🙂
Cheers,
George