I have been studying galois theory on my own and find it very fascinating. I have gone through Ian Stewarts book: http://www.amazon.co.uk/Galois-Theory-Third-Chapman-Mathematics/dp/1584883936. I am not sure which book to study from next but I would like to learn about something along the lines of algebraic closures of Q and other related topics.

Related to this, I would also like to learn more finite field theory. I would like books that are not too technical and provide some context since I am doing a self study.

Field Arithmeticby Fried and Jarden contains quite a bit of (infinite) Galois theory. Furthermore,Algebraic Patchingby Jarden is mostly about a relatively recent technique in Galois theory. – Peter Mueller Apr 3 '13 at 13:24