There are several things one can learn when going into homotopy theory.
Since I don't know how much you already know about these kinds of topics I will include many possible things you could do.
On the one hand I think it is worthwhile to get familiar with simplicial sets (do you know them?) and their relationship with topological spaces.
A good (In my view from the modern language the best) reference is "Simplicial Homotopy Theory" by Goerss and Jardine.
When studying the relationship to topological spaces now it depends on how much homotopy theory you already know.
You should learn about CW complexes, fibrations, homotopy groups, the Hurewicz theorem, the Whitehead theorem, the long exact sequence of a fibration, the factorization via the cylinder and the homotopy fiber, CW approximations and Postnikow-sections.
I suppose all of this is treated at least in the book "Elements of homotopy theory" by G.W. Whitehead. Propably there are also newer treatments of this, for example "Modern Classical homotopy theory" by J. Strom. I don't know this book myself, but a good friend of mine has read in it.
Then what you could learn next, I think, would be the language of model categories.
A good introduction is the paper by Dwyer and Spalinski called "Homotopy theory and model categories" ( here is a link to a pdf http://folk.uio.no/paularne/SUPh05/DS.pdf )
Then in the book of Goerss and Jardine in chapters 1 and 2 you will learn a lot about this and the importance of simplicial sets in this business.
It is worthwile to learn many examples of model categories (e.g. Top with Quillen model structure, Top with Strom model structure, Chain complexes, simplicial sets, sequential spectra...)
After all of this the next thing to lean into could be foundations in stable homotopy theory. But for the moment I think this is enough already.
If you tell me what you already know, maybe I could give a more precise suggestion of what to learn next, as I figure what I've told you is about 2 or 3 courses and a seminar :)
Hope it helps a little bit,
Best regards
Best Answer
Homotopy theory / algebraic topology was born out of applications rather than abstract nonsense considerations. So there's plenty of applications, as that's how the subject began.
Perhaps the first topological proof would be the bridges of Konigsberg problem: http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg
Where algebraic topology started getting off the ground was in the work of Poincare. The Poincare-Hopf Index theorem: http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_theorem
was a landmark. In its natural setting it was a relationship between Euler characteristic, tangent bundles and intersection theory. But from the perspective of a differential equator it's a fundamental tool that allows you to determine whether or not differential equations have fixed points.
Applications have piled-up over the years. Some of the more modern ones are listed in other people's responses. The birth of topological dynamics in the mid 20-th century was of course a big one.