1. Topology by James Munkres.

In my opinion this books is a definitive one in point-set topology, it contains all the basics that everyone should know, and it is written in a magnificent way, I dont have to say that it was one of my favorite books to study. It is a very readable book and is very friendly for self study. It has very illuminating examples and great motivation for some of the most important theorems. I recommend to study all the first part on point set topology, except maybe, the last two chapters (ch 7 and ch 8)

1. Topology by James Munkres.

Again, I think that this book is the best reference to learn for the first time this topic. I recommend to study in detail all the second part on algebraic topology. You can give a quick read to chapters 10 and 12, just to get the big picture of the important theorems.

2. Algebraic Topology by Allen Hatcher.

This one is a standard reference in many algebraic topology courses. It is a great book that covers almost all the basics of algebraic topology. Although it is a great book, sometimes I dont like the  way that it is written. At some points I feel like it should be more concise and that there are some examples that are not suited for a first time study. Hatcher postpone some proofs or results after a brief discusion of examples, motivation or lemmas that he will need in the proof, and sometimes he gives you a proof while he is giving you an intuitive motivation of an idea, so if you skip some page or paragraph, you would never know if you missed something important until some pages later. 

Besides all its cons it is still a great book, one of the biggest pros of this book (for me) is that it has a lot of problems to work on, so I would recommend to supplement Munkre's book with this one.

1. Elements of Algebraic Topology by James Munkres.

What can I say, I am a big fan of Munkres haha. I think this book is excellent for a first time study, however it has a major con for me and it is its part on simplicial homology, so I recommend to skip all three first chapters and start directly in chapter 4 on singular homology. As before it would be a good idea to supplement this one with Hatcher's book (chapter 2 and 3, with sections 3.A and 3.B of the additional topics). 

1. Introduction to Smooth Manifolds by John M. Lee (First edition).

Every topologist (algebraic or not) must know the basics about smooth manifolds. This books is excellent for self study, it is very clear and concise. I would recommend to study chapters 1 to 16. You can ommit chapter 9.

1. Category Theory in Context by Emily Riehl.

At this point, I would recommend to learn about category theory. This book is very readable and has many (many) examples, maybe that is its biggest con, in a first read you won't know which examples you should skip, however I think it would be enough if you stick to the first two or three examples that she uses to illustrate each concept. A full read of this book would be good, if you want to skip something for later maybe chapters 5 and 6 can be ommited. I recommend to put special attention to the rest of the chapters, they are a daily tool for any algebraic topologist.

2. Categories for the Working Mathematician by Saunders Mac Lane.

This book is a standard reference for the subject. I used this book to learn category theory for the first time and it is good, but I have to say that it is a little bit dry and terse. I would recommend to read the first 10 chapters.

1.  Algebraic Topology by Tammo Tom Dieck

For me, this is a great book and one of the things that I like the most about it is the order of the topics. It is written in a very clear way, it is concise and goes direct to the point, however its point of view is very categorical and formal so it can be hard to follow, and at some points you will lack the geometric intuition behind the concepts if you stick only to this book. The first three chapters are worthy to read again because it introduces the fundamental group as a especial case of the fundamental grupoid, it will add another angle to your perspective!. Another con of this book is the short amount of problems. I recommend to read chapters 4, 5, 6, and 8 very carefully.

2. A Concise Course in Algebraic Topology by J.P. May.

In my opinion if this book wasn't so concise it would be the perfect book. It presents the basics that everyone should know and nothing more. It is a great book to come back later, after a first study of the subject. It has an enormous advantage that is implicit in its brevity: it will tell you at which theorems pay special attention, so you can use it to supplement Tammo's book in order to know at which theorems you should focus.  You should read chapters 6 to 11.

3. Algebraic Topology by Allen Hatcher.

This is a great book to turn to when you feel lost reading Tammo's book. You can use it as a complement, reading section 4.1 and 4.2 with Tammo's book. It will give you the geometric intuition that you will need to develop the right ideas about the constructions (something that Tammo's book lacks), however I feel that through these sections the content is a little bit disorganized, you will find mentions to fibrations here and there, and also to fiber and cofiber exact sequences and other constructions. Section 4.3 is much better and I think it is a must read. Also, for a first reading you should study sections 4.A, 4.E and 4.F of the additional topics.

1. Vector Bundles and K-Theory by Allen Hatcher.

A very readable book for a first time introduction to K-theory and Vector Bundles. It is written very clearly and concise. The first two chapters are enough to learn the basics of this subject.

2. K-theory by M.F. Atiyah.

This is a classic reference, although it is quite old (it was written in the 60's, now almost 60 years ago, can you believe it?) However, it is a good place to learn about K-theory. I would recommend to read it after Hatcher's, or use it as a principal reference and turn to Hatcher's book when you feel lost.

1.  Fiber Bundles by Dale Husemöller.

Not my favorite book but is a good place to learn the basics of fiber bundles and principal G-bundles. Its biggest con, I think, is that it approaches the subject with the greatest possible generality, however, you can be pragmatic and stick to the case when the base space is paracompact, everything will be simpler. I would recommend to give a good read to chapters 2 and 4 for a first glance of the subject. 

2. Topology of Fiber Bundles (Lecture notes) by Ralph Cohen.

If you like something more concise and direct to the point these are great lecture notes to read, it can be a good complement to Husemöller's book. Here is the link to his notes. I would recommend to put special enphasis in chapter 2, specially sections 1, 2, 3 and 4 (This last one is very important!).

Also, chapter 3 will work as a good introduction to characteristic classes.

2(1/2). At this point it will be useful to learn about Thom's isomorphism theorem and the Gysin sequence. Section 4.D of Hatcher's Algebraic Topology is an excellent place to look at. You should give a detailed read to this section!.

3. Characteristic Classes by John W. Milnor and James D. Stasheff.

This is a classical reference to learn about characteristic classes, however, it can be difficult to follow some parts if you dont have certain tools (for example they prove the existence of Whitney-Stiefel classes using Steenrod square operations), so I would recommend to give it a try after the references that I gave before. This book is worthy to read completely, if you have troubles with any part I would recommend to give it a quick read to get the main ideas and keep moving foward. 

1. 1. Spectral Sequences by Allen Hatcher.

Hatcher has some notes on Spectral Sequences in his webpage. The first version are three separated pdf that were intended to be a book in spectral sequences, these are dedicated to the Serre sequence, the Adams sequence and the Eilenberg-Moore sequence. The most recent version is a single pdf that is intended to be a continuation to his Algebraic Topology book, and it is more o less the same as the 3 pdf before together. It is a good introduction to spectral sequences and it has the pro of being full with many examples and applications in topology, but also lacks a little bit of the algebraic background that one needs to feel comfortable using spectral sequences, thus my recommendation is to commplement it with McCleary's book.

1. 2. A User's Guide to Spectral Sequences by John McCleary.

This is a very friendly book for the student, it is almost like a "spectral sequence for dummies". It biggest con can be that sometimes it goes too slow, however, spectral sequences are a very messy tool that every topologist should master, so it is better to take it slowly and to make sure you really understand how it works and how to use them, and this is basically one of the goals of this book. The autor himself indicates the way you should read the book, and which specific sections you should read if you want the fastest and painless way to understand spectral sequences, also he has marked some sections that are not intended for a first read so you can skip them.  My recommendation is to complemment the general background on spectral sequences that you will need from this book, so you can read it together with Hatcher's book.