I agree that your goals are relevant to this question.  I.e. do you want to "learn", "understand", or "use" algebraic geometry?, or perhaps write a thesis?  These are all different.  If one wants to understand the subject, I like the historical approach, beginning with Riemann surfaces over the complex numbers, say from a book like that of Rick Miranda (augmented by reading Riemann).  I.e. I think it is useful when learning an abstract subject to know what elementary things it generalizes, rather than just memorizing the general version.

For varieties it helps to supplement Mumford's red book by Shafarevich's Basic Algebraic Geometry.

It is also my view that if you want to use the subject in geometry, including calculate with it, that sheaf cohomology is more important than schemes.  Thus George Kempf's book (or Serre's FAC) which treats cohomology of varieties, may be more useful than studying schemes.  if you want to do number theory on the other hand, I am assured that schemes are fundamental.

I guess the historical order would be roughly: Riemann surfaces, algebraic curves, algebraic surfaces, general projective varieties, sheaf cohomology of varieties, schemes and their cohomology.....

If you want to learn Riemann surfaces and sheaf cohomology at the same time, Gunning's Princeton lectures on Riemann surfaces are excellent.  But there you will only learn the analysis and not the geometry.


I am surely hopelessly naive, but to me schemes are just varieties where you also remember the equations, and stacks are just moduli spaces where you remember the automorphism groups.