I am interested in learning the basics of group theory. I know a few basic theorems like Lagrange, Cauchy etc., but nothing from group actions. What would you recommend?
Mathematicians generally fall into two broad categories of intellectual style: geometers and algebraicists. (The greatest mathematicians spanned both styles.) Those of us geometers prefer to think visually and spatially, while algebraicists prefer symbols and formalism. Some branches of mathematics are best approached by one or the other of these cognitive styles, e.g., Number theory for the algebraicists, and Differential geometry for the geometers.
Group theory is an interesting branch of mathematics in which both styles can thrive. If you're like me (a geometer), then by far the best book on group theory is: Visual Group Theory by Nathan Carter.
I'll let an algebraicist suggest a book favoring that cognitive style.