How set Tags And Groups field set to NOT "collapsed on display"
To add on to @oskylark answer I would use an extra template instead of a duplicate. Easier to manage when upgrading. https://civicrm.org/blog/dave-greenberg/now-its-easier-add-custom-behaviors-templates

In the extra template (TagsAndGroups.extra.tpl) and you could throw some javascript in a {literal} tag along the lines of

cj('.crm-tagGroup-accordion').collapse('toggle');


NOTE: I did not test this js line, just a first guess.