Make a full back up of the site including:
All web site files on the server
The database in full
Upgrade to the latest WordPress version ensuring that the site design and functionality stays intact
Repairing of anything broken in the theme if needed after upgrade
Updating of all plugins to the latest version

Additional item needed: Updated server php version to a newer version for website WordPress compatibility
Removed unused plugins and themes for hardened security
Secondary complete back up of web site after upgrade including:
All web site files on the server
The upgraded database in full