Algebraic Geometry for Topologists As someone who is


*

*familiar with algebraic topology, say, at the level of Hatcher's book, and

*familiar with homological algebra and categories and applications in topology

*but has no idea what a variety is


what is a good place to start learning algebraic geometry?
 A: (I guess my opinion is no more worthy of being an answer than the opinions in the comments, but it's verbose, so let me put it in the answer box anyway.)
As a beginning PhD student I knew a reasonable amount of algebraic topology, similar to what you describe in the question. But I don't think it really gave me any extra or better choices in how to start learning algebraic geometry. As Steven Landsburg's comment suggests, the kind of objects one studies and the methods one uses in algebraic geometry are so much more specialised than arbitrary (even nice) topological spaces that you really need to start from scratch, with something like Shafarevich (as suggested by Mark Grant). 
That isn't to say that having a good knowledge of algebraic topology won't be useful to you in learning algebraic geometry --- quite the opposite, in fact. (Example: characteristic classes.) And, as you progress in algebraic geometry, you will likely run into more and more topics where your topology knowledge gives you a great head-start on understanding. But it won't really help with those first steps.
On the other hand, if I had to nominate a beginning algebraic geometry textbook that is oriented towards the topological point of view, I guess it would be Principles of Algebraic Geometry by Griffiths and Harris. But, great resource though it is, I don't think I can recommend that book to anyone in good conscience as a first introduction to algebraic geometry. 
A: A similar question was asked while ago, which was made a community wiki later on. You may find it useful, if not exactly what you would have expected. Here is 
A learning roadmap for algebraic geometry
