How do I change the styling of the post tags under the posts?

Use the following CSS in your Customizer -> CSS or your style.css to add a tag icon in front of each tag:

.tag-groups-premium-post-tags-table td a {
  text-decoration: none;

.tag-groups-premium-post-tags-table td a:before {
  content: "\f323";
  font-family: Dashicons;
  margin: 0 .2em 0 .4em;
  padding-right: .1em;
  padding-left: .5em;

The output will be like this:

Similarly, you can add other styling. If you are not comfortable with using CSS, your web designer will be able to help.