Given a group how to investigate groups that are homomorphic to it

This is a more general question related to another thread I just posted.

Given a group G I am relatively comfortable looking for other groups which might be isomorphic to it. I'm not great at it, but I have a few ideas I can check out. Doing the same with homomorphisms is giving me a headache.

There is, of course, the trivial homomorphism (ie the one that maps every element of G to the identity), and I'm pretty sure there is homomorphism between a group and any of its subgroups but what does one do when confronted with two groups G and H? As a negative example of what I am asking, how would I show that two groups are not homomorphic (beyond a trivial homomorphism, that is.)

To look for isomorphisms I know to check the number of elements, check to see if there are the same number of subgroups of the same size, etc. ie. the group structures must be the same. That's relatively easy. Knowing that the two groups are candidates I can then go on to construct an isomorphism.

How do you approach this problem using homomorphisms? Or am I barking up the wrong tree here?

Thanks!

-Dan