We have the calendar as we want it on the currently live site.
We are changing hosting providers, and in turn redesigning the site on a new server. We would like to move/export the calendar into the new site, without having to re-type all the events.
That’s an excellent question. I would suggest using the WordPress Export Tool on the current site (Tools > Export), then using the WordPress Import Tool on the new site (Tools > Import). Events are an offshoot of posts, so they are compatible with those tools just as posts are.