How to add "Read More" on Tags page

I have a tags page that gets a lot of traffic, but people leave this page quickly because they don’t click on the title of the page to go read the full version of the actual page. They just see the abbreviated text on the tags page. Is there a way to add a “Read More” link just after the abbreviated text so they can click on this link?

Also, what template or file does the tags page use? I tried looking for this file/template and could not find it.

Thanks in advance.