Assuming that you had an agonist and an antagonist for every autonomic transmitter receptor, how could you determine which receptor types exist in any autonomically controlled effector?

The correct answer is by using the antagonist.

The antagonist is a molecule that blocks a biological response by binding to the receptor. So, you add antagonists to the receptors you want to determine and see which antagonist blocked the response. By blocking the specific response you can get the answer what receptor it was.