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 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.