Homology & covering spaces

I am looking at a question for a practice class tomorrow (2.2.12 from Hatcher):

Show that the quotient map $\displaystyle S^1\times S^1 \to S^2$ collapsing the subspace $\displaystyle S^1 \vee S^1$ to a point is not nullhomotopic by showing that it induces an isomorphism on $\displaystyle H^2$. On the other hand, show via covering spaces that any map $\displaystyle S^2 \to S^1\vee S^1$ is nullhomotopic.

For the first part, I'm not sure what the induced map looks like, so I can't use it to form an isomorphism; for the second, I'm unsure how to use the covering space to show that the map is nullhomotopic.