Another book worth looking at is Differential Geometry and Lie Groups for Physicists by Marian Fecko,
https://www.amazon.com/dp/0521845076/?tag=pfamazon01-20.
This book is not as rigorous as the books by Lee and Tu, but it more rigorous and comprehensive than the book by Schutz. Fecko treats linear connections and associated curvature, and connections and curvature for bundles. Consequently, Fecko can be used for a more in-depth treatment of the math underlying both GR and gauge field theories than traditionally is presented in physics courses.
Fecko has an unusual format. From its Preface,
A specific feature of this book is its strong emphasis on developing the general theory through a large number of simple exercises (more than a thousand of them), in which the reader analyzes "in a hands-on fashion" various details of a "theory" as well as plenty of concrete examples (the proof of the pudding is in the eating).
The book is reviewed at the Canadian Association of Physicists website,
http://www.cap.ca/BRMS/Reviews/Rev857_554.pdf.
From the review
There are no problems at the end of each chapter, but that's because by the time you reached the end of the chapter, you feel like you've done your homework already, proving or solving every little numbered exercise, of which there can be between one and half a dozen per page. Fortunately, each chapter ends with a summary and a list of relevant equations, with references back to the text. ...
A somewhat idiosyncratic flavour of this text is reflected in the numbering: there are no numbered equations, it's the exercises that are numbered, and referred to later.
Personal observations based on my limited experience with my copy of the book:
1) often very clear, but sometimes a bit unclear;
2) some examples of mathematical imprecision/looseness, but these examples are not more densely distributed than in, say, Nakahara;
3) the simple examples are often effective.