Just to add to what was said here ...
Git is actually really good for this sort of thing. You can update Moodle using Git, if you installed Moodle using Git. https://docs.moodle.org/35/en/Git_for_Administrators
Most plugins also have a Git repo, so you could do those with Git too ... but I would stick to doing plugins via Moodle.org and the Plugins directory, because at least there the developer has released the plugin as stable for the Moodle versions it is tagged for.
So the ideal process would be to update Moodle with git, and update plugins using your Moodle site's plugin install/upgrade feature.
To get there from here however you would need to:
- clone your database and data directory
- install Moodle afresh from Git
- copy in all those plugins
- update the config.php to use the cloned DB and cloned data dir
- run the Moodle upgrade
- check nothing broke and fix up anything that did
Then you could either use the new site as is, and turn off the old one. Or repeat the process for the production site now that you know it works.