If you would like to learn algebraic topology very well, then I think that you will need to learn some point-set topology. I would recommend you to read chapters 2-3 of Topology: A First Course by James Munkres for the elements of point-set topology. If you would like to learn algebraic topology as soon as possible, then you should perhaps read this text selectively. In particular, I would recommend you to focus mainly on the following (fundamental) notions, reading more if time permits:
- Topological space
- Basis for a topology
- The product topology (on a finite cartesian product; if time permits, you can read about the case of an infinite cartesian product but this is not urgently needed as far as algebraic topology is concerned)
- Subspace topology
- Closed set and limit point
- Continuous function
- Metric space
- Quotient topology (this is a very important, but sometimes ignored, prerequisite for algebraic topology)
- Connected space
- Component and path component
- Compact space
- Hausdorff space
- The separation axioms, Urysohn's lemma and the Tietze extension theorem (if time permits; these are very useful and inteteresting concepts but you can take the Urysohn lemma and Tietze extension theorem on faith if you desire)
I think that as far as algebraic topology is concerned, there are two options that I would recommend: Elements of Algebraic Topology by James Munkres or chapter 8 onwards of Topology: A First Course by James Munkres. The latter reference is very good if you wish to learn more about the fundamental group. However, the former reference is nearly 450 pages in length and provides a fairly detailed account of homology and cohomology. I really enjoyed reading Elements of Algebraic Topology by James Munkres and would highly recommend it. In particular, I think a good plan would be:
- Learn the elements of point-set topology as outlined above.
- Read chapter 8 of Munkres' Topology: A First Course to learn the rudiments of the fundamental group.
- Read Elements of Algebraic Topology by James Munkres.
You will not need to know anything about manifolds to read Elements of Algebraic Topology but I believe that it is good to at least concurrently learn about them as you learn algebraic topology; the two subjects complement each other very well. I think a very good textbook for the theory of differentiable manifolds is An Introduction to Differentiable Manifolds and Riemannian Geometry by William Boothby (but this is a matter of personal taste; there are (obviously) many other excellent textbooks on this subject). The advantage of this textbook from the point of view of this question is that there is a flavor of algebraic topology present in one of the chapters.
I hope this helps!
I will respond directly to this part of your question.
I would be happy if I could avoid such topics [analysis], but I don't know what type of mathematics is studied in a graduate degree level, so this leads me to the following questions:
What topics of Abstract Algebra should I study in depth ? what topics in Abstract Algebra should I be familiar with their basics ?
Are there any topics in analysis, topology etc' that are likely to be needed for answering a graduate degree level type of questions ?
What should be the focus of my work, should I try to do many exercises within the text, or focus on the proofs and the theory ?
Are there topics in Abstract Algebra, or other in other areas that I would need to know (maybe topology ?) that I can skip some parts of (mainly non-core topics that are hard to learn) since they would probably not help me (and due to lack of time) ?
I have the book Abstract Algebra by Dummit and Foote to study with, as well as books in other area of mathematics such as Topology by Monkers that might help me with this goal.
Firstly, I want to mention that unless you are absolutely certain that you are going to specialize in pure group or ring theory, then you will need some analysis. In fact, you'll probably need a lot of analysis. Explaining why is a bit more complicated. The short version is that almost every area of math relies or is at least informed by analysis, algebra, and topology; this is why most graduate programs (in the US anyway) require these as either graduate classes or graduate entrance exams or graduate qualification tests, etc.
To expand in a slightly longer way - calculus is pretty interesting, and lets you do a lot of things. A common thing that mathematicians do is put measures on weirder spaces so that you can have some variant of integration. In number theory (even algebraic number theory, which is often the same thing as algebraic geometry, which is often the same thing as commutative algebra, which is just algebra and group theory), we really like having measures called Haar measures on matrix groups, like the $GL(n), SL(n), Symp(n)$, etc. This lets us do integration on these groups. So we study functions invariant under actions of these groups, or functions invariant on certain cosets of these groups that behave nicely under ring translation, or some similar idea. And one way we do this is to integrate them, or consider an integral over a weighted average of a function across the cosets our function is invariant over (read: Eisenstein Series for example), to extract largely algebraic information about number fields. Or we consider representations (as in representation theory, which I clump into the larger algebra domain sometimes) and analytic extensions of representations. Everything I've mentioned here requires a certain comfort with topology, analysis, and algebra.
This is to say that algebra mixes quite a bit with analysis in many ways. You would really benefit from having a good understanding of analysis and topology. In particular, don't focus solely on algebra. The other answer says this a little, but I am going to emphasize this a lot. It is very important to understand analysis and topology, unless you are going do limit yourself to pure, remote group theory. And even then, I wouldn't recommend it.
But back to your question at hand about algebra:
I would prescribe a path into algebra. In a comment on the other answer, you mention that you know groups, ring, fields, Galois theory. Cool! You also say you have Dummit and Foote (by far my preferred introduction to group and ring theory). Then I suggest two paths:
Go learn more about whatever parts you liked most. Sylow theorems interest you? Try to learn your way through Burnside's theorem. You like Galois theory? Pick up some infinite-dimensional Galois theory and try your hand. Maybe you already know that? Go pick up some algebraic number theory text - as an intro algebraic number theory text builds nicely on basic field theory and Galois theory, and suggests further paths. To be fair, I'm biased - I'm a number theorist. The important thing is that you go and dig deeper into things that interest you.
Pick up Atiyah and MacDonald's Commutative Algebra (hopefully from a library, as they're proud), and do your best at all the exercises. This is the 'natural' extension of what to do next, and it's the real path into a serious interest into algebra in my opinion. I say that you should do all the exercises because this book is famous for having really important lemmas and theorems in the exercises as opposed to the exposition. This will also really set your group theory and ring theory in stone, and you have Dummit and Foote to fall back on if you need. If you know this already, you should next go to Lang's Algebra (quite a bit, scary thing - take a look at it first), Matsumura's Commutative Ring Theory (much, much, much higher than Atiyah MacDonald, even though they have essentially the same name), or Eisenbud's Commutative Algebra (also harder than Atiyah MacDonald, but designed for people interested in algebraic geometry - if you don't know what that is, look it up).
I'd like to add one more thing about your (3) - the problem with learning the proofs and theory is that there is no reason for them to stick on their own. You might open up Atiyah MacDonald and understand everything you read, for example. But I wouldn't expect much of it to last, unless you use it. So a good general philosophy is to read and try to absorb, but then do exercises to let it solidify. Well written exercises require you to build on the text, both as a review and to build intuition.
A hard problem is knowing how many exercises to do. Too many, you waste your time. Too few, you'll forget much. But this is sort of moot, as it's hard to know what problems are useful or good to do before you actually do them, and in some texts some problems are much much better for you than others. For this, I advise you to ask your advisor (or find someone who can provide some sort of guidance) for direction once you have an idea what sort of things you want to learn about.
Best Answer
For algebra, Dummit and Foote is as easy and gentle an introduction as you could ask for. It's not quite as easy for topology.
I happen to write a blog post that collected various answers and topology book recommendations from MSE. For ease, I've copied over the content below. But to follow any hyperlinks, which I didn't copy over (Wordpress has annoying formatting), I'd recommend just looking at the post itself or googling the corresponding term.
I learned from Herstein for algebra and Munkres for topology, and they both require you to do exercises to grasp the material. And that's how learning almost anything else in math from a text will be, so that's okay.