I like Thurston's treatment in his book. The idea is that any finite subgroup $G< SU(2) \to SO(3)$ gives rise to an orbifold $S^2/G$. First, one classifies the possible quotient orbifolds, then one figures out the possible preimage subgroups in $SU(2)$. Exercise 4.4.6 gives a direct argument (at least for $SO(3)$). The lengthier, but more conceptual argument using orbifolds does not appear in the published book, but is in section 5.5 of a preliminary draft (presumably this would be part of the material to appear in volume 2), and also appears in Theorem 13.3.6 of Thurston's notes. Classifying spherical and euclidean orbifolds is a satisfying exercise, that may be undertaken by undergraduates with very little mathematical background.

I like Thurston's treatment in his book. The idea is that any finite subgroup $G< SU(2) \to SO(3)$ gives rise to an orbifold $S^2/G$. First, one classifies the possible quotient orbifolds, then one figures out the possible preimage subgroups in $SU(2)$. Exercise 4.4.6 gives a direct argument (at least for $SO(3)$). The lengthier, but more conceptual argument using orbifolds does not appear in the published book, but is in section 5.5 of a preliminary draft (presumably this would be part of the material to appear in volume 2), and also appears in Theorem 13.3.6 of Thurston's notes. Classifying spherical and euclidean 2-dimensional orbifolds is a satisfying exercise, that may be undertaken by undergraduates with very little mathematical background: see the notes from the course "Geometry and the Imagination".