If there exists an isomorphism between two groups, then the two groups are exactly the same up to relabeling of elements (and change of binary operation) - the phrase "preserves group structure" is generally used to say exactly that. An analogy would be like telling a story with the names of people changed like in /r/relationships - we get the story and the names are pretty much irrelevant. We say the two groups are isomorphic if an isomorphism exists.
If two groups has generating elements that correspond to each other (equal number of elements of the corresponding order) then they are isomorphic. Similarly, if one group has an element of order k and the other one doesn't, then they are not isomorphic (no corresponding element). These are what I would probably check first when given a group I'm unfamiliar with in assignments.
Homomorphisms don't say much by themselves, but there exists theorems that helps you extract some information from the domain group.
There are even more special types of isomorphisms called automorphisms (isomorphisms that maps a group to itself). These pop up a lot in Galois theory courses, but I don't know enough of the topic to comment anymore.