How do I use and show a vocabulary with multiple fields?
What you're looking for is Entity Reference.  This will let you create a taxonomy term field on your content type and give you the option to render the entire term using the display of your choice (teaser, full etc).

Setting it up would be very similar to what you've already tried with term reference:


Create an entity reference field in your content type
Target type = taxonomy term
Set target bundle if preferred, then save
Under manage display, set the field format to rendered entity
Set the view mode to your preferred option (you would configure these options under your vocabulary manage display screen)