John Stillwell's book Naive Lie Theory motivates the subject in the first chapter using an exploration of quaternions acting by conjugation on the unit sphere. I think his explanation is great for someone at any point in their undergraduate career, however the subsequent chapters may be extremely slow for anyone already acquainted with lie theory, group theory or general topology.
http://www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathematics/dp/0387782141
There is a much more in depth coverage of the details of this in another book which I am trying to recall the name of.
edit: the book Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli by Toth begins with a chapter on platonic solids and finite rotation groups all very explicitly, in terms of Möbius transformations. Once you understand how to interchange between Möbius transforms and quaternions acting on $R^3$ from stillwell, this may be helpful.
In 3D space, initial coordinates are decided by the given Left-hand and Right-hand coordinates, sometimes called x-up, y-up, z-up conventions which can also be found (for example) in Wiki's EulerAngles item.
Initial Quaternions are defined corresponding to the intial x-y-z
For example, x-y-z-right hand coordinates, then 0,0,0,1 would be the initial quaternion.
Of course, the order also depends on the setting of program.
For example maybe DirectX and openGL are in different orders of the initial quaternion of the same coordinates.
And also your self programs could in one arbitrary initial order.
However, when one initial order was defined(corresponding to x-y-z), then the other 23 orders could also be well defined.
4!
(*
24
*)
24 is the number of kinds of orders. When I decide which order was used in one 3D game engine which I cannot know the order by specification, I do experiments.
Maybe you could find a Matrix to Quaternion program, and then do rotations in Matrix, then convert the combined final matrix to Quaternions will lead you a good comprehension.
Best Answer
The place to look is John Baez's beautifully written article on The Octonions. The introduction is wonderfully entertaining and the relevant section you want to focus on is the Cayley-Dickson construction.