The 'devils are in the details'! So far we know what the job is and desired goal ... what we don't know are the 'devils'!
Platform? (OS) Are you averse to using command line?
So the quickie ... 2 ways:
Migrate the entire site then march the moodle upwards.
Stand up a 4.highest server, setup categories that are the same as old server, restore course backups.
One of the things you would want to do ... on old server know the plugins you have installed and check Moodle plugins site to see if those plugins have a compat version for your destination version of Moodle. Themes are plugins.
References:
https://moodledev.io/general/releases
Using above resource see what the requirements are for higher versions of Moodle core.
https://www.syndrega.ch/blog/#php-and-dbms-compatibility-of-major-moodle-releases
https://docs.moodle.org/500/en/Moodle_migration
https://docs.moodle.org/500/en/Git_for_Administrators
Note: the version numbers in the URL's above are the same docs one would use for your version of Moodle.
'SoS', Ken