Getting started with Lie Groups

I would suggest you start with chapter 4 of An Introduction to Manifolds by Tu, Then study Lie Groups, Lie Algebras, and Representations: An Elementary Introduction by Hall and finally study Differential Geometry, Lie Groups, and Symmetric Spaces by Helgason.

Good luck!

Stillwell: Naive Lie Theory is a great first introduction since it covers the very basics and uses SO(3) and SU(2) as examples. And it gives a sneak-preview of group theory and topology too. Though, more advanced topics such as adjoint representation and left-invariance are not covered.

The book of Zhelobenko is written with applications in view . It has a whole chapter on $SU(2)$ (which is isomorphic with $SO(3)$), so it seems to fit with the paper you're interested in.