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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, $\text{Spec }A$ where $A$ is a finitely generated algebra over some field is 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 out there: 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?

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?