There are numerous benefits to running your moodle site using HTTPS. This increases the level of security especially involving sessions and passwords.
Steps
Before you start
Check all the content you use supports https. Use the HTTPS conversion tool to do this.
Make sure you have a staging environment. You will want to set up HTTPS the first time on a staging environment rather than updating your live site. It will take some time to convert to https and you will need to update content (see below).
Setting up an SSL certificate
The first thing you will need to do is acquire an SSL certificate. You can create these yourself, but this is only helpful for development purposes. Instead you will want to get your SSL certificate from a certificate authority, so that the certificate will be publicly verified.
The cost of certificates has been somewhat prohibitive, they come at various costs from a few dollars to hundreds of dollars per year. For the budget constrained, the "price is right" with a new initiative brought to us by the Internet Security Research Group (ISRG). Free domain-validated certificates can be acquired from Let's Encrypt. Let's Encrypt also tries to make the process of installing and managing certificates as painless as possible and there are numerous methods and clients available.
Setting up your server
Then you will need enable SSL on your web server to add your certificate. This process will vary depending on your web server of choice.
If you are using a proxy or load balancer, depending on your setup you will most likely want to set up the SSL certificate on your proxy server
Setting up your Moodle
On a basic Moodle site, it will be simple to set up https. Simply edit config.php and change http:// to https:// in $CFG->wwwroot.
However, if you are using a proxy or load balancer, depending on your setup you may need to set $CFG->sslproxy to 1, and not use SSL on the Moodle server. Then the load balancer or proxy server can communicate directly to your Moodle site, but serve to the clients over SSL.
Redirecting the HTTP address to the new HTTPS address
You may want to add a redirection so that users who have the current http:// address to your site will be redirected automatically to the https:// address instead of getting a page with a message such as "For security reasons only https connections are allowed, sorry."
There are several ways to do redirections, depending on your web server, hosting, and how you prefer to do this. A common solution on Apache web servers is to create an .htaccess file with a rewrite rule such as:
RewriteEngine On
RewriteCond %{SERVER_PORT} 80
RewriteRule ^(.*)$ https://www.yourmoodle.com/$1 [R,L]
or in general
RewriteEngine On
RewriteCond %{SERVER_PORT} 80
RewriteRule ^(.*)$ {SERVER_NAME}/$1 [R,L]
Be sure to look up current ways to do this for your web server and host, such as the how-to article Force HTTPS connections in an Apache server environment.
Updating content
You will need to change all embeded content from being requested over http. Links do not matter. But you will need to update images and iframes, scorm modules, and LTI external tools. You can modify external tools to open in a new window instead of in an iframe and they will work fine.
A new tool was added to Moodle 3.4 to aid in this process. This is available via a link in Site administration > Security > HTTP security > HTTPS conversion tool. See MDL-46269 for details.
