Self teaching Galois Theory
Solution 1:
A good book to self study from, since it contains solutions to all the problems, is Abstract Algebra: The Basic Graduate Year by Robert Ash, available for free here. You can read about the style in which he writes his books here.
Solution 2:
I would jump straight in at the deep end. Try Stewart. I haven't read his Galois theory book, but his other books are very accessible, so you should manage on your own. You can Google online lecture notes for pretty much any background material you need.