As another commenter mentioned, currently Ravi Vakil is teaching a course in modern algbraic geometry.
The course is largely based on Vakil's book in progress, which he titled The Rising Sea, after a quote from Grothendieck:
> I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! . . .
> A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance.
Here is, roughly speaking, a theorem that illustrates what modern algebraic geometry is good for.
Theorem. If you have two conic sections in the plane, which don't share a common component, then they intersect in exactly four points.
Here a "conic section" is just any quadratic equation in two variables, e.g. x^2 + y^2 + 3xy + 4x = 1. "Common component" means that the conic sections are identical or overlap in an entire line.
Now, here are some "examples".
Concentric circles, e.g. x^2 + y^2 = 1 and x^2 + y^2 = 4. They don't intersect at all, right? Ah, except you forgot to count points over the complex numbers, where they do.
Tangency, e.g. y = x^2 and x^2 + (y - 1)^2 = 1. These curves are tangent at (0, 0), so you have to count this point with multiplicity.
One major goal of modern algebraic geometry is to wrap these sorts of considerations into the foundations. So you have to work a lot harder to even say what a conic section is, or what it means for two of them to intersect. But, once you've laid the foundations in this manner, there are no special cases.