You can craft your Class Browser URL with multiple tags in it so the class browser loads with them automatically checked and filtered when the page loads.
For multiple tags without group reference, the URL structure is https://[siteurl]/class/browse?tagId,tagId&matchtype=1. The matchtype is optional. If you leave it off, Match All will be checked. If you include it, use 0 for Match All and 1 for Match Any.
For a tag in a specific group and a tag in any group, the URL structure is https://[siteurl]/class/browse?groupIdtag,tagId. Use the group Id for the specific group in which you want the tag to be checked.