This has been a big week but peacockmedia.software is now https://
This is a well-overdue move. Google have been offering small carrots for a long time, but at the end of this month, they'll be adding a stick as well. They're switching from informing users when a connection is secure, to warning users if a connection is insecure. Google Chrome is making this move but other browsers are expected to follow suit.
Well-informed web users will know whether they really need a connection to be secure or not, but I suspect that even for those users, when this change really takes hold, the red unlocked padlock will start to become an indicator of an amateur or untrustworthy site.
Once the certificate is installed (which I won't go into) then you must weed out links to your http:// pages and pages that have 'mixed' or 'insecure' content, ie references to images, css, js and other files which are http://.
Scrutiny makes it easy to find these.
1. Find links to http pages and pages with insecure content.
First you have to make sure that you're giving your https:// address as your starting url, and make sure that these two boxes are ticked in your settings,
After running a scan, Scrutiny will offer to show you these issues,
You'll have to fix-and-rescan until there's nothing reported. (When you make certain fixes, that will reveal new pages to Scrutiny for testing).
2. Fix broken links and images
Once those are fixed, there may be some broken links and broken images to fix too (I was copying stuff onto a new server and chose to only copy what was needed. There are inevitably things that you miss...) Scrutiny will report these and make them easy to find.
3. Submit to Google.
Scrutiny can also generate the xml sitemap for you, listing your new pages (and images and pdf files too if you want).
Apparently Google treats the https:// version of your site as a separate 'property' in its Search Console (was Google Webmaster Tools). So you'll have to add the https:// site as a new property and upload the new sitemap.
[update 15 Jul] I uploaded my sitemap on Jul 13, it was processed on Jul 14.
As part of the migration process, Google recommends that you then "Redirect your users and search engines to the HTTPS page or resource with server-side 301 HTTP redirects" (full article here)