Varieties as an introduction to algebraic geometry / How do professional algebraic geometers think about varieties  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.
 A: 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!
A: I stand by my answer to the question that you linked.   In particular, I think that the distinction between "classical" and "modern" algebraic geometry is a little artificial, and I don't think that anyone is meant to do any particular thing; 
what you need to know depends on what theorems you want to read/use/prove.
But whatever direction you ultimately intend to pursue, it makes good sense to learn
varieties first.  As well as Chapter I of Mumford, there is Chapter I of Hartshorne, and
its many exercises.  The first few sections in particular are crucial.  There is also
Griffiths and Harris, which has no mention of schemes, as far as I can recall, but an awful lot of algebraic geometry of varieties.
A: I don't seem to have the option to edit my answer, but I wanted to correct myself: it is the scheme structure of the limit that determines the possible nearby varieties in the family, not vice versa.  The question of whether the nearby varieties determine the limit is that of Hausdorfness of the parameter space.  One has to restrict the nature of the possible limits to get uniqueness of the limit.  This arises in deciding just how complicated the limit should be when trying to construct a nice compactification of a given family of nice varieties.  It is a wonderful fact that for smooth curves one can compactify them without introducing non varieties.  All one needs is some simple singular curves.  This was first noticed by Alan Mayer and David Mumford in their talks at the Woods Hole conference 1964, whose notes appear on James Milne's webpage at Michigan, (and mine at UGA).  I am roy smith, and have been using mathwonk as alias for so many years online I have forgotten it is not my name.  Is it sufficient to register and add it to my profile, or do I need to sign posts here as roy smith?  That would apparently unlink me from all my mathwonk activity so far.
A: 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.
A: 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.
Of course everyone is different, as my friend George Kempf apparently just sat down and read EGA, but that didn't work for me.
For varieties it helps to supplement Mumford's red book by Shafarevich's Basic Algebraic Geometry.  Joe Harris's book Algebraic Geometry derives from his experience teaching algebraic geometry first by concrete examples at Harvard and Brown, but very little theory, which he said seemed to work well.
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.
A remark: one way to think of schemes is as a "limit of varieties".  This is not quite general, but if one has some equations (f1,..,fr) in n variables, they give a map from n space to r space, whose fiber over most points is usually a smooth variety, but whose fiber over the point 0 is more special.  That fiber, with its scheme structure, essentially determines the (possible) nearby varieties that may be thought of as converging to the special fiber over 0.  Thus knowing the  scheme structure can help tell you how the object would change if its structure is slightly varied.  E.g. the scheme defined by x^2=0 tells you it is a limit of two points, while that defined by x=0 is a limit of one point.
