An excellent book for the beginners is
MR0917939MR0917939 Nikulin, V. V.; Shafarevich, I. R. Geometries and groups. Translated from the Russian by M. Reid. Universitext. Springer-Verlag, Berlin, 1987.
EDIT: And there is a book based on Arnold's own lectures:
MR2110624 AlekseevMR2110624 Alekseev, V. B. Abel's theorem in problems and solutions. Based on the lectures of Professor V. I. Arnold. With a preface and an appendix by Arnold and an appendix by A. Khovanskii. Kluwer Academic Publishers, Dordrecht, 2004.