For algebraic geometry there are a number of excellent books. Hartshorne's Algebraic Geometry is widely lauded as the best book from which to learn the modern Grothendeick reformulation of Algebraic Geometry, based on his Éléments de géométrie algébrique. This is also, however, considered one of the most challenging textbooks ever written on any mathematical subject ever. This makes it a poor first choice. He leaves critical and very much non trivial results as exercises and has very few motivating examples outside of the exercises. But it is well worth the effort.
Others are The Red Book of Varieties and Schemes by Momford which covers the topic from a more classical viewpoint. This is an excellent book. Also from the classical and historical view is Basic Algebaric Geometry I by Shafarevich which is well written and also an excellent book.
I would strongly reccoment reading one of the last two to get a grounding in the historical motivation and classical theory, then read Hartshorne. Be warned though, reading Hartshorne may, and should, take years.
There is an excellent book by Allen Hatcher called Algebraic Topology that is available for free on his website, and also as a hard copy on Amazon. This is an excellent geometrically oriented book on the subject that contains much of what you would learn in a graduate course on the subject plus a large number of additional topics. This is where I reccommend you first go.
A second excellent text on the subject is Topology and Geometry by Bredon. This book isn't titled as if it an Angebraic topological work but it is, and it is particularly well written, although requires a bit of maturity and knowledge of category theory.
Finally we have J.P. May's A Concise Cource in Algebraic Topology. This is probably the best text for a second exposure to the subject, or to be used along with one of the two I have just described. It doesn't give a lot of exaples and proceeds very quickly through the subject (hence concise) but it places the field in context and provides motivation for many of the objects of study. This one comes highly recommended as well.