This part assumes that you are upgrading to a version that is already made available as a package by the system administrators.

1. Go to MySite and select your site;
2. Click Package Add > MediaWiki xxx version;
3. You will then see the MediaWiki versions listed, select your intended version to upgrade, then click install;
4. As shown in the phabricator issue https://phabricator.wikimedia.org/T311966 , you need to remove these lines in order to load the website properly:

$path = array( $IP, "$IP/includes", "$IP/languages" );
set_include_path( implode( PATH_SEPARATOR, $path ) . PATH_SEPARATOR . get_include_path() );
require_once( "$IP/includes/DefaultSettings.php" );

5. Upgrade the extensions and skins. These extensions can be get from the MediaWiki website.

6. Run the update script. Either update by typing http://your-domain/xxupdate.php or http://your-domain/mw-config. I personally recommend using the second one as preferred by MediaWiki officially.

With the above steps, the upgrade is launched fully. It should take for 30-60 mins for the upgrade, so make sure you leave enough time for the upgrade and notify the users to save their edits.