There are two things to check/do before upgrading from 3.0.19.x to a higher version of Event Espresso 3:
1. Back up your database. There are a lot of plugins that can do this. We recommend WP DB Backup. Having a backup of the database means you can go back to the way your site was before if something goes awry. This is also a good basic practice to follow when updating your website plugins, themes, or making change to the operating system(s) of your website. Get in the habit of backing up your website.
2. Make sure you do not have any currently active events. The issues we have had reported from earlier versions of Event Espresso have all been surrounding attendee data and payment information. We have not been able to reproduce the problems on our test servers, but we are handling and calling some data differently in the newer versions of the plugin, so the problems that were reported were around those areas. But our recommendation is still the same. Schedule time to update your plugins when your events are not actively accepting registrations.
If you have any questions or problems, then please create a support post in our support forums so we can help you.
We recommend migrating to Event Espresso 4 when the time is right for you. If you have an active support license, you will have access to the EE3 plugins that will help you upgrade to a more recent version of Event Espresso, which will also allow you to upgrade to Event Espresso 4.