Once you have your GTM account container created you are ready to add your first tag.  This will usually be the Google Analytics tag.

From the Workspace menu tab, click the New button to create a new tag.  This will launch a page that allows you to create a tag and its corresponding trigger.  From the Choose Tag Type menu, select Universal Analytics.  The page will then present several configuration options.  To simply launch analytics, add your GA tracking ID (either as the value itself or as a GTM variable) and set the Track Type to Page View.  You can leave the other configurations alone for this tag since all we want it to do is launch the base GA code.  On the Trigger section, select All Pages.  Save this tag/trigger combination.  Now any time a page loads the GTM code it will execute the GA code.

In order to run both your own and the university Tracking IDs, simply create one tag for each.  If you do this, also be sure to go into the More Settings on the tag configuration and in the Fields to Set add allowLinker – true and cookieDomain  auto.

One easy way to see that your configuration is working is to go into Preview mode.  Select the pulldown menu in the top-right that says Publish and change this to Preview.  Now open a new browser tab and visit the page that contains your GTM code.  The panel at the bottom of the page will show the tags which are firing and not firing.