I can highly recommend "A Book of Abstract Algebra", by Charles C. Pinter. You'll learn about groups, rings and fields. You will also learn enough Galois Theory to understand why polynomials of degree higher than $4$ are, in general, not solvable by radicals.
It is 'formal' in the sense that it is rigorous, but the author is also very good at explaining the intuition behind all ideas. It is much less dense than most Abstract Algebra books, and, in my opinion, and excellent introduction to the subject.
Furthermore, it isn't expensive and it contains solutions to numerous exercises. See the amazon page of the this book for more positive reviews. Again, highly recommended!
Added: once you finished this book, you're ready for more advanced treatments of abstract algebra. After Pinter's book, you could try "A First Course in Abstract Algebra" by John B. Fraleigh. After that one, a great option is "Abstract Algebra" by Dummit and Foote. This is quite an advanced textbook, but a good one nevertheless. Once you've worked your way through these books (I advise you not to just read through them, but actually soak up the information by doing the exercises and reading actively) you will have a strong basis of knowledge in abstract algebra. By then you can tackle more advanced topics.