How much of scheme theory can you visualize? I am just starting to learn about schemes and algebraic geometry in general, but I am finding it very hard to visualize things. For example, affine schemes that look like varieties are easily visualized. But how about infinite dimensional or nonproper things? Or fibre products of schemes? So just throwing out this question to all algebraic geometers: When you are doing your research, how much of your results come from the geometric intuition? If one were to start the research in algebraic geometry, what would you say the most important things is?
 A: I decided to post this separately:
an excellent blog post by Lieven Le Bruyn.
In this post, he analyzes in detail Mumford's drawing (in the red book) of the "arithmetic surface".
Just wonderful!
A: I've found this set of notes http://math.mit.edu/~poonen/papers/Qpoints.pdf by Bjorn Poonen very helpful in learning arithmetic geometry. I'd read everything very carefully and do all the exercises there. Bjorn is a master of exposition.
A: Well, you asked 10 different questions, and I am not sure what you mean by "nonproper" ($Spec A$ is not proper). But let's see.
A scheme is a very geometric object, with practice - or maybe just habit - one learns to visualize it quite well. If you already see geometrically $Spec$ of a finitely generated algebra over a field $k$ (including algebras with nilpotents which you visualize as "thickenings", including $k$ not algebraically closed which you visualize as Galois orbits; you looked at these, right? these are important steps) then you are almost there. Add some other standard examples such as $Spec(\mathbb Z)$, DVR, a double-headed snake (the first nonseparated scheme), and you already know plenty do start doing research.
Infinite-dimensional algebras? Well, I suppose it is just as hard or easy to imagine them as infinite-dimensional spaces.
The fiber product is a perfectly geometric notion as well, and fairly easy to visualize. You begin by looking at fiber products of sets and you progress from there through some standard examples. Isolating a fiber of a morphism is an important case. And then look at some examples where the residue fields of the scheme points change. Learn the simple way to compute the tensor product $A\otimes_R B$ by using generators and relations of $A$, and you will be up and running in no time.
As far as the balance of geometry vs algebra, I suppose that depends on a person and everybody is different. My advisor used to say that geometry comes first and then later algebra follows, and I tend to agree. I think you get nowhere without geometric intuition.
But if you are serious, at some point you will need a solid commutative algebra foundation. Fortunately these days there are plenty of nice books, starting with the very nice and elementary "Undergraduate commutative algebra" by Miles Reid.
A: I suggest looking at the illustrations in The Geometry of Schemes by Eisenbud and Harris and Algebraic Geometry by Hartshorne.  Both books have many carefully drawn pictures of schemes that are not varieties.  In general, I think it helps to compute, or at least have in mind, some relatively simple examples of a phenomenon.  For example, if you want to view the graph of a morphism as a fiber product over the diagonal of the target, you can try the complex affine line first.
I don't think most people make much of an effort to visualize infinite dimensional schemes, or even complicated finite dimensional schemes.
A: It depends on what you mean by "visualize"...
For things like properness, separatedness, finiteness, smoothness, fiber products, tangent vectors, etc., I usually found it helpful to first understand the analogous things for sets and/or (real, differentiable) manifolds. But if you do this, you have to also keep in mind that the situation for schemes is usually more subtle than the situation for sets and manifolds; while the situations are analogous, they are sometimes not exactly analogous. Another issue is that you might not always know what the appropriate differential geometry analogues of many algebraic geometry concepts are; they're not always spelled out in commonly used books like Hartshorne. For example, I spent a very long time being totally confused by the valuative criterion for properness and could never remember the statement --- until I realized that properness corresponds to compactness in differential geometry, and that the valuative criterion is roughly analogous to sequential compactness (which is equivalent to compactness, for metric spaces, e.g. manifolds) in general topology.
Another thing that I find helpful is to first understand things in the complex setting before moving to the general setting. For example, complex varieties are examples of complex manifolds; complex curves are just Riemann surfaces. The best way, in my opinion, to get a basic handle on for instance finite morphisms is to understand what they are in the case of complex curves: they are branched covers of Riemann surfaces.

Aside: Personally, I think that students of algebraic geometry should take a course on Riemann surfaces before (or while) taking a course on abstract algebraic geometry. Certainly I wish that I had done this. The theory of Riemann surfaces is very beautiful and I think it really helps to motivate some of the "big theorems" that you typically build up to in a first course in abstract algebraic geometry: for example Riemann-Roch and Riemann-Hurwitz.
Probably it is not hard to guess that I also think that students of algebraic geometry should have taken a course on differential geometry before taking a course on abstract algebraic geometry.
A: One thing that the other answers haven't addressed yet is infinite-dimensionality. But every algebra over a noetherian ring is a filtered colimit of finitely generated algebras, which can be visualized individually. For example, $\mathbf{C}(z)$ is the colimit of the rings $\mathbf{C}[z][1/f(z)]$, as $f(z)$ runs over more and more highly divisible nonzero polynomials. So it is reasonable to view $\mathrm{Spec}\ \mathbf{C}(z)$ as the limit of the affine line punctured by an exhaustively increasing family of points (closed!).
In fact, if I remember the category of schemes [edit: quasi-compact and quasi-separated] is equivalent to the category of projective systems of schemes of finite type whose transition maps are affine. So, in some basic sense, the finite-dimensional case determines everything.
The main thing I've never been able to find a way to visualize is the Frobenius map in positive characteristic. 
