This really is two questions, but they are kind of related so I would like to ask them at the same time.
Question 1:
In a question asked by Amitesh Datta, BCnrd commented that it is important to learn about varieties in a classical sense before learning about modern algebraic geometry because it is where much of the intuition in the subject comes from.
I was hoping to get some opinions on how much one should learn about varieties (in the sense of chapter 1 of Mumford's red book) before moving onto more modern formulations of algebraic geometry.
Is one meant to gain a rudimentary understanding of varieties and then start learning about schemes, OR is one meant to have a really good understanding of abstract varieties before learning about schemes.
Question 2:
Do professional algebraic geometers think about varieties from a scheme theoretic perspective or from a classical perspective.
This is a seriously soft question, so I will make it community wiki. I am however half expecting it to be closed.
Best Answer
When you are truly fluent in scheme theory, you don't know whether you are "thinking schemes" or "thinking varieties", the intuitions are merged together.
As to learning, for most people starting with schemes is a bad idea, because they don't get to build the necessary intuition, and unmotivated formalism can be quite repulsive; but there are (very few) students with unusually abstract inclinations for whom starting with schemes is just fine.