What kind of recommendations do you have for a very good source to learn Galois Theory? Is there any Atiyah-MacDonald-type book on Galois theory? What is your opinion on the chapters from Lang and Dummit and Foote?
I really enjoyed learning Galois theory from Martin Isaacs' Algebra: A Graduate Course. Isaacs' textbook is a textbook on group theory, ring theory, and field theory (in other words, algebra!) so it's not just on Galois theory. However, you'll have a very complete knowledge of Galois theory if you read the latter half of the textbook where it is discussed. The textbook also has the distinct advantage of good, challenging exercises. The emphasis of the exercises in this textbook is on theory more than on specific computations and examples (although these are discussed as well; Isaacs generally feels, I suspect, that a student reading his textbook is already quite comfortable with specific examples and computations so should be able to do them independently). If you'd like to see worked computations and examples in detail, then perhaps it is a good idea to supplement Isaacs' textbook with textbooks like the one by Dummit and Foote on the same topic.
My suggestion would be Exploratory Galois Theory by John Swallow. This develops the basic theory that one would find in any course in abstract algebra, but from a very concrete perspective, so it seems easier to understand on a first read than other textbooks. See link
Arnold has some lectures on Abel's theorem that you may find interesting. See Abel's Theorem in Problems and Solutions: Based on the lectures of Professor V.I. Arnold by V.B. Alekseev.