Textbooks for learning Algebraic Topology

I guess the Fulton (Algebraic Topology, a first course) would be a good choice. He stays quite elementary throughout the book, and there are hints for most exercices at the end.

Elementary Topology Problem Textbook by Viro, Harlamov, etc. as an introduction It covers only part of a subject, but it has solutions.

Check out From Calculus to Cohomology by Madsen. Here is the link

http://www.amazon.com/From-Calculus-Cohomology-Characteristic-Classes/dp/0521589568

It is at a lower level than Munkres and has some good simple examples. All the exercises are at the end of the book. Unfortunately there are no solutions, but there are a fair amount of hints.

Also Introduction to Topological Manifolds by Lee has some nice material in it.

http://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1461427908/ref=sr_1_2?s=books&ie=UTF8&qid=1458733673&sr=1-2&keywords=introduction+to+topological+manifolds

When I learned the subject I found the background I learned from Lee's book very helpful when I read Munkres. Once again no solutions, but the authors exposition is second to none.