Unique carbons on rings In rings in general, how do I determine how many unique carbons there are? In this example, I am thinking that there are 2 unique carbons (one on top and one with a double bond) since this molecule is symmetrical. Any feedback is appreciated.
Yes, one would try to find points or lines of symmetry in general to check for equivalent ring atoms. It immensely helps to have a model and a lot of practice.

In your specific example, notice that you have two lines of symmetry in your (idealized) diagram: one passing through the sp3 carbons, and one crossing the two double bonds. You can thus "fold" the diagram into "quarters", leaving you with two non-equivalent atoms.