I study math this way, and many people I know do--I think most graduate students shift to studying math this way since once you have a basic foundation built up, problem driven learning is better (since now you don't need to know what's on the test--you need to know enough to succeed in research, and so you need to make sure you learn valuable tools that come up often).
You mention the Weil conjectures; they might be ambitious for a first course in algebraic geometry. As a first attempt to learn, say, the material in Hartshorne, I think it would serve one well to try and understand the following problems.
1. Let k be a field. How do you classify nonsingular projective curves over k? This is a great question, since it forces one to grapple with how to define 'nonsingular curve' over a general field. I imagine one would first want to construct all \*proper\* curves over k, and then learn about linear systems to see how to construct a projective embedding of a proper curve.
2. Prove the Weil conjectures for algebraic curves--Weil did this using clever but classical arguments.