Subgroups of finitely generated groups are not necessarily finitely generated
No. The example given on Wikipedia is that the free group $F_2$ contains a subgroup generated by $y^n x y^{-n}, n \ge 1$, which is free on countably many generators.
It is well-known that the free group $F_2$ on two generators has as a subgroup a group isomorphic to a free group on a countably infinite set of generators. See Qiaochu's example.
However a finite index subgroup of a finitely generated group is finitely generated.