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.