Hi John
You'll need to publish a page which will be your homepage, to which you will need to add two Views, or this is a good example of how you can recycle the same View.
So, create a new View which shows your Production posts.
Add a Query Filter for your taxonomy, and when you insert it specify that the term should be set by a shortcode attribute.
You'll want to specify a limit option of, say, the most recent 4 posts. The default order options match your requirement.
The Loop Output section is where you design the output. You should use the Loop Wizard and specify Bootstrap grid for the format, with, say, 4 columns, whatever makes sense for the output style you want, then include the fields you want to appear for each result, and gather these in a Content Template.
Now, when Views creates the template it includes just enough markup/styling to display the output in a grid. You will need to intervene editing the markup and CSS to have it look exactly how you want.
Next, go to your front page and edit it.
Use the Fields and Views button to insert the wpv-view shortcode where you want the first group of results.
With the dialog that appears, choose the Query Filters tab, and set the term "auditorium" for the first set of results.
Then repeat this to insert the same View a second time, this time using the "studio" term.
As I say, you will need to expend a certain effort in styling the results to your taste, but the functionality should give you what you are aiming for.
Let me know if you get stuck.