References on principal G bundle and connections I am trying to understand about principal G bundle given a Lie group $G$. For that I started with action of Lie groups on manifold $M$ and convinced my self that if the action is smooth, proper, free then orbit space is a smooth manifold and corresponding projection map gives a fiber bundle $(M,\pi, M/G)$. We call this bundle and any other bundle isomorphi to this bundle a principal $G$ bundle. Only to understand just this it took so much time for me as there are many books with different definitions of principal bundles.
I am comfortable with this definition of principal bundle and would like to know more  enough to start reading connections on principal bundles. So, I would like to ask for your suggestion regarding some reading materials online on principal bundles. I checked it but did not find satisfactory. Any suggestion regarding references are welcome.
Please do not add husemoller fiber bundles as it uses some other definition of principal bundle and not Kobayashi also. I am benifited very much with chapter on Lie group actions on manifolds in Lee’s Book Introduction to manifold. I wish he had published some notes on principal bundles as well. Please do let me know any references along this lines.
Edit : I am now able to understand roughly the concepts of connections, holonomy groups from Kobayashi. Any reference which can supplement this book is most welcome. Now I am reading curvature form and structure equation and do not really understand what they are saying. 
 A: "Foundations of Differential Geometry" by Kobayashi and Nomizu has a good introduction to principal bundles and connections on them.  In fact, this is presented at the beginning and is used as the basis for their presentation of classical (in the present) differential geometry. 
A: Try the book by Michor, Kolar & Slovak titled Natural Operations in Differential Geometry, it's rather dense so if you're uncomfortable with Kobayashi & Nomizu you might find it doesn't work for you. It has a couple of chapters on vector, fibre, principal and jet bundles. It's also available as a free download. It's very comprehensive and best treated as a reference text. 
Another book to try is Saunders The Geometry of Jet Bundles, this is more accessible but the notation is very heavy; he goes up to the variational bicomplex.
Also Moritas The Geometry of Differential Forms, I haven't read this apart from a brief look at its contents, but it looks very interesting.  
A: I realised one can not (should not) escape from reading Kobayashi and Nomizu’s book.
Other books that helped me to learn more about principal bundles are


*

*Differential Geometry: Connections, Curvature, and Characteristic Classes by Loring W. Tu

*Connections, Curvature, and Cohomology Volumes 1,2,3 by Werner Hildbert Greub, Stephen Halperin, James S. Vanstone, Ray Vanstone

A: I like Chern's short and easy Vector bundles with a connection, in Chern, Global Differential Geometry, M.A.A., 1989. Chern focuses on vector bundles but does so by using simple examples of principal bundles. Chern gives some serious theorems that motivate learning the subject.  
