How to compare 2 groups to see if they are equivalent under isomorphism? What is a productive way to find wether wether 2 groups are equivalent classes under isomorphism. 
Examples: 
- Klein Four V4 and Z3
- Dihedral group of a square D4 and Symmetric group S3
Thanks
 A: Your question is, in some sense, the central question of abstract group theory. It is an extremely broad question which there is no straightforward answer to. But there is a general strategy which can be formulated:


*

*To prove two groups $G,H$ are isomorphic, construct an isomorphism. To put it another way, construct an appropriate function $f : G \to H$ and prove that $f$ is an isomorphism.

*To prove two groups $G,H$ are not isomorphic, find some isomorphism invariant of groups and show that $G,H$ have different values for that invariant. For example, the cardinality of a group is an invariant under isomorphisms: if $G,H$ are isomorphic then they have the same number of elements; and so you could prove that $G,H$ are not isomorphic by proving that they have different numbers of elements. (Hint: this should be useful for the examples in your question).

A: If you are asked to prove/disprove that two groups are isomorphic and you suspect that they are not isomorphic, looking at the orders of each element of each group can be helpful. If two group are isomorphic, then they possess the same number of elements for any given order. This is nice if the groups are relatively small. 
A: 
What is a productive way to find whether 2 groups are equivalent classes under isomorphism. 

Understand the groups, and group theory, and mathematics, and use software and literature, and ask people, and ...
There is no computer algorithm to do what you ask for two arbitrary groups that are presented in some way (it is a famous undecidable problem).   For groups that have additional restrictions or come from settings with extra structure, one can use that information to find isomorphisms identifying the groups, or invariants that distinguish them. For instance, if the groups are symmetry groups of some structured objects, use the structure on the objects to draw some conclusions.  The objects may not be of group-theoretic nature themselves, so one has to use knowledge of other mathematics and use it to constrain the group theory.
