I'm having some problems understanding what elements are in a quotient ring. I understand that they are cosets of the ideal, but when it comes to actual calculations I'm still a bit lost.
For example, consider the ring $\mathbb{Q} [x,y,z]/(x,y) $. Am I correct if I think that the quotient essentially set all terms containing $x$ or $y$ to $0$, so that we end up with simply $\mathbb{Q}[z]$ (or something isomorphic to it)?
It is right, but be carefully (although not being this the case) if you have algebraic relations between elements and the ones you view as $0$ these relations involve changes on those elements. In this case the situation is simple as the elements $x,y,z$ are (algebraically) independent. Think at your ideal as a set of algebraic relations which you impose on your ring, getting the quotient with your imposed relations (and their consequences, i.e. other elements in the ideal)