Polyatomics -How Do They Work?
Our teacher started with $\ce{AuCl}$ and wrote down Gold (I) Chloride as the nomenclature. How did he get from $\ce{AuCl}$ to Gold (I)?
There are two possibilities: Gold (I) and Gold (III).

True, but there is only one possibility for chloride ;)
Karl's comment under your question gives the crucial hint:
Start with the part that you know and that is unambiguous!
Let's have a look at some examples:

Chloride is (always) $\ce{Cl-}$; it has a charge of (-1).
If your teacher would have written $\ce{AuCl3}$, you would know that there are 3 chloride ions with a total charge of $3\times(-1)=-3$. Consequently, you would know that you're having $\ce{Au(III)}$ here. The copper (II) cloride example belongs to the same category and if you like, you can try iron(II) bromide, iron(III) chloride and copper(I) iodide.

Karl also mentioned oxygen. You can safely assume one or more $\ce{O^{2-}}$ units in your metal oxides, depending on the oxidation state of the metal. Can you assign the oxidation state in $\ce{Fe2O3}$?

Nitrite, nitrate, sulfite, sulfate are some anions that you should memorize sooner or later. You might know that the names that start with nitr~ refer to nitrogen, the sulf~ refers to sulfur. All these anions contain various number of oxygen atoms.