In addition to Mr. V's and Howard's advice ... both semi repeated here ...
Lot's of planning ahead.
Highest version of 3.8 is 3.8.9 (Build: 20210510)
If you plan to march the site upwards, first order of business is to do a full
site backup and store on server. That's code + DB dump + minimally moodledata/filedir/
Before doing anything with the site, set the theme to boost via config.php file
and leave it like that for the duration of the march.
$CFG->theme='boost';
You can worry about how site looks after arriving at destination version .. for the march concentrate on function ... not form.
Get the 3.8 site under git versioning ... side load or upload ... and get to the 3.8 to highest.
One reason for doing that ... scripts in admin/cli/ .. one of which is uninstall_plugins.php
which will help you ID additional plugins installed in the current site.
Git makes marching much easier ... just for core moodle.
I'd also highly recommend the installation of moosh - command line utility for Moodle.
A version compatible with version of 3.8.
https://moodle.org/plugins/pluginversions.php?id=522
0.39 (2021032400)
Note: you will have to upgrade moosh as well.
Plan the march checking Mr. V's chart and Moodle releases IO
https://moodledev.io/general/releases
making note of required PHP version, MySQL/MariaDB version + php extensions.
You can't get those higher than what is supported moodle version wise.
Now some questions:
How are you hosted?
Does hosting offer attached data devices that you can detach and re-attach to another server?
How are users currently authenticated?
Does site make use of any external resources that would require re-config at the other end?
There is an alternative to a site march. Individual full course backups.
XFer those to a file system repo on new server. Restore using moosh or scripts in admin/cli/
Catch 22 for that approach - add-on plugins in current site .. do they exist in destination version?
Lot's of planning and fun and games ahead! 
Hope your 'customer' doesn't expect this to happen in a short time frame!
'SoS', Ken