When you do a Facebook or Google share, you may want your featured image to be used as a share thumbnail. You simply need to install and activate WordPress SEO or Jetpack. In Jetpack, you will need to make sure you activate the share module. In WordPress SEO, you need to make sure you have checked the option to display your Open Graph metadata.
These plugins insert the correct meta tags into your pages necessary for social media sites. Here is what sharing looks like after installing one of these plugins.
If you have already tried to share your post before you set your featured image, or activated sharing, then Facebook has cached your share. So any new attempts to share your post will show the same old result. Go to Facebook’s Open Graph Object Debugger, and clear your share cache.