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.