Hi there!
You're absolutely correct, you would just need a domain-level server redirect to take all http URLs to their https equivalent.
Best practice is to ensure you've also got the non-www. version of the website covered in that same redirect too, to avoid any chains. You'll see what I mean if you run your domain through this > https://varvy.com/tools/redirects/
Depending on what stack you're using, here are the 2 guides. One for htaccess and one for IIS:
You shouldn't need to, but just to be on the safe side, I would also add canonical tags to the http pages, pointing to their https equivalent prior to putting the server level redirect in place. This is to ensure that you won't be causing yourself issues if the redirect fails for any reason. Details here:
Once you've got your redirection planned in, make sure you set up a Google Search Console account for the https version to ensure there are no crawl issues and to check that the http version of the site stops receiving traffic.
That should just about cover it!
Hope it all goes well,
Sean