study presentations, zum beispiel:
$\langle a,b\mid a^2=b^3=e, ab=b^2a \rangle$
vs
$\langle a,b\mid a^2=b^3=e, ab=ba \rangle$ or $\langle a\mid a^6=e \rangle$
compare nuances...
Other: the $Out\pi_1(F)$ of any surface. Btw this is the famous mapping class group of the surface $\cal{MCG}(F)$\cal{MCG}(F)$.
To get a real modern grasp on the subjet you should include topological techniques: homotopy, cohomotopy, homology and cohomology, K-theory... you might enter in contact with a types of structures like free groups, free products, amalgamated products, HNN extensions of groups, Grothendieck groups and the like...
