How do you know what metal is an anode/cathode?

The best way to figure out which metal is an anode and which is a cathode is to look at them chemical equation that is provided to you. Something my chemistry teacher taught me was this:

Red Cat  and An Ox  translated the element being Reduced is the Cathode and the one being Oxidized is the Anode.  

If your unaware of how to determine which is oxidized and which is reduced my class was also taught:

Leo the lion goes Ger - in this one your looking at the first and last words,  Losing Electrons = Oxidation, Gaining Electrons = Reduction.

The reason why zinc is an anode and Copper a cathode is because you're talking about a battery.

The more reactive metal tends to undergo oxidation (loss of electrons) more easily, and their electrons will travel to the other metal. Because of this the more reactive metal is normally the positive terminal, or the anode.

Zinc is more reactive than copper, so zinc is the anode and copper the cathode.

well reduction occur in the cathode and oxidation occur in the anode. Since zinc is more electropositive than copper. Therefore zinc undergoes oxidation that is losses electron making it the anode while copper behave like an electronegative element to undergo reduction making it the cathode.