1 · Export before you touch anything
Export your events as .ics from the current plugin while it is still installed and active. Most calendar plugins can do this; if yours cannot, look for a WordPress export of the event post type, and keep a database backup regardless.
Do this first because plugin data is often only readable through the plugin's own code. Deactivate it first and you may be looking at rows you cannot interpret.
2 · Read the export
Drop the file into the reader above. Three things to check.
Are recurring events still rules? If your fortnightly quiz exported as fifty separate entries, the export flattened the series. You can import that, but you will get fifty events instead of one and you will be editing them individually forever. Consider re-creating those few series by hand instead.
Did cancellations survive? Look for exclusions. If you cancelled dates and they are not in the file, they will reappear on the new site.
Do events have timezones? If they exported as floating times, they will be interpreted in whatever zone the importer prefers, and you will have an hour's error across the whole calendar.
3 · Import into a staging copy
Not the live site. Import into a copy, with both plugins installed, and compare. Look particularly at a few months of every recurring event — that is where importers differ most, and where an error is least visible.
4 · Plan the URLs
This is the part people forget, and it is the part with commercial consequences. Your event URLs are in Facebook posts, newsletters, printed flyers and Google's index. A new plugin will use a different URL structure.
Before switching, list your existing event URLs — an SEO plugin or your sitemap will give you them — and set up redirects to the new ones. For a venue, an old listing that 404s is a customer who was coming to something and now is not.
5 · Switch, and keep the old one for a fortnight
Deactivate the old plugin rather than deleting it. Leave it a couple of weeks. If something is missing, its data is still there and reactivating is a click.
Delete it only once you are sure, and read what its uninstall routine does first — some plugins remove all their data on deletion, which is fine if you meant it and a catastrophe if you did not. A plugin that keeps your data by default and asks before removing it is the safer neighbour.