What examples to use when learning group theory? I am trying to learn group theory from the textbook Algebra by Michael Artin 2nd edition. I am looking for finite group and subgroup examples that are useful for developing an intuition about group theory. So far, I have been using the symmetric group $S_3$. 
Preferably, I am looking for an example of a finite group and a subgroup that I can use to grasp the concepts of the book. A list would be nice. Or just direction towards a database of group examples could work too.
 A: Of course any answer to such a broad question has to be incomplete, but you might find the following helpful:

At the very beginning I think that (as you say) $S_3$ is a good example as an easy-to-grasp nonabelian group. And abelian groups aren't entirely uninteresting either - you should understand why $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}\cong\mathbb{Z}/6\mathbb{Z}$ but $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\not\cong\mathbb{Z}/4\mathbb{Z}$. This sets the stage for the classification of finite(ly generated) abelian groups, and ultimately of finitely generated modules over a PID, but long before then is just a key point to understand - and the latter also serves as a good example of how the "obvious" statement $$A/B=C\implies A=B\times C$$ fails miserably in the context of groups, quotient groups, and direct products.
What about a bit further on - e.g. when we start talking about normal subgroups? In my opinion, $A_4$ and $A_5$ are quite good examples. Each is non-abelian, and $A_5$ is simple while $A_4$ is not - and that's already a good thing to have examples of. And proving that $A_5$ is the smallest non-abelian simple group is a good problem. And of course, eventually you'll care about solvable groups, and then $A_5$ is crucial.
There are also examples of groups which early on will just seem weird, but are really valuable. For example, a small semidirect product like $D_3$ (yes, that's just $S_3$, but thinking about it as a dihedral group is a bit better right now). It's easy enough to write out how this group works (without defining semidirect products - just defining this specific group), and compare it with the corresponding direct product $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\cong\mathbb{Z}/6\mathbb{Z}$; while it will then be a while (probably) before you care about semidirect products, it will be useful to have in your head early on the idea that there are more complicated ways to combine groups than just the direct product.
And then there are infinite groups which are still "finite-flavored" - I'm thinking e.g. of matrix groups like $GL_n(\mathbb{R})$, the group of all invertible $n$-by-$n$ matrices with real number entries. This isn't such a great example if you're not already familiar with some linear algebra, but if you are it's really quite good to get your hands on it early.
A: Yes, you will have to learn the basics of algebra systems. 
Magma Calculator
http://magma.maths.usyd.edu.au/calc/
and (or) Wolfram Mathematica. Magma is best to generally investigate the group structure. Mathematica is useful to support Magma with symbolic calculations and edge cases, when Magma is not convenient.
Making paper tetrahedrons and other regular n-gons for rotation symmetries will be a cool experience as well.
