I believe the right answer to this question depends very much on how quickly one needs to get up to speed on schemes (and at this point in mathematical history, stacks).
For me personally, I found learning about complex varieties (from Mumford's book "Algebraic Geometry I: Complex Projective Varieties" and Griffiths-Harris) much more entertaining than learning the basics of scheme theory. This is largely a matter of personal taste, but for some it's already a good enough reason to start with varieties: if it's just more fun for you to work with particular examples, varieties are the place to start.
On the other hand, I didn't have to write a thesis in algebraic geometry and it wasn't until later on in my career that I needed anything about schemes--at which point I had the luxury of plenty of practical experience with varieties in characteristic zero to rely on. I can't overstate how much I relied on my experience with algebraic varieties when I began learning about schemes: I could appreciate the added flexibility schemes provide and I already had a zoo of examples under my belt. Without earlier experience with varieties I would have found myself yawning through the mountains of routine but necessary scheme-preliminaries.
But schemes are really an indispensable part of modern mathematical language if you are in one of the many fields that rely on algebraic geometry, and it may therefore be essential to learn about them immediately in parallel with varieties. I don't envy the modern graduate student who has to decide on a tipping point between classical algebraic varieties and schemes/stacks!