How (and why) learn algebraic geometry: UW walkthrough

1. Before one begins, one needs to have a solid knowledge of commutative algebra, which supplies the "local models" for the spaces. A course such as is mandatory. For homological algebra, one can in principle get away without doing it, but the course Methods of algebra in geometry and topology is very helpful: in particular it supplies basics of sheaves. Once one has the two courses, the formal definitions in algebraic geometry are fairly natural.
2. The course Algebraic Geometry and the seminar Foundations of algebraic geometry are complementing each other. As a rule of thumb, each can be taken before each other and doing the second one is much easier. In principle, one could only do one, but it makes sense to do both for practice. I have a slight preference on doing Foundations first. Foundations use a beautiful and freely available book by Ravi Vakil. Other possible sources include the book by Gortz and Webhorn (slightly too dry on the first run, truly excellent for experts), the book by Hartshorne (classical, yet with a habit of hiding tons of technicalities in the exercises) and others. The website Stacks project is excellent and likely the LLMs will serve as a good bookkeeping device for it (although, as of today, google still does a better job).
3. The gap between intuition and formalism in algebraic geometry is huge. One needs to master both. The bonus — and it is a huge one — is twofold. First, once one acquires some (not perfect!) understanding of the field, it is possible to work in many directions from applications to complexity theory, to arithmetic geometry, to algebraic topology and homotopy theory. Second, once one passes the initial shock of generality, many many works start to "make sense": the field is very much interconnected. Still, to understand what algebraic geometry is, it is best to talk to someone at MIMUW; the group is large.