Google have long been pushing the move to https. Browsers now display an "insecure" message if your site isn't https://
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. If you started at an https:// url, and you had the above boxes checked, then you'll automatically see this box (if there are any issues).
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)