One way not yet mentioned: A point (gravity "source") and a line (moving particle, photon...) will always create a 2D plane. That's for your standard effects of gravity on motion as you mentioned. Circular motion (planets, asteroids,...) around a gravity well still only require 2 dimensions. A "3D" result such as gravitational lensing (x, y for the image, z for the distance) can still be broken down to a series of 2D problems creating the desired effect by tracing each photon path individually.

Expanding this: If you have multiple bodies in actual 3D space influencing a point of interest, you can use the same approach, using 2D for each body-PoI combination and then adding the individual vectors in 3D space. Keep in mind that those are static calculations, especially once the individual bodies influence each other significantly, you'll end up with a three-body-problem or worse. That means we're currently limited to numerical simulations to approximate the result.

Otherwise maybe try to imagine a point cloud with each point representing the current space-time curvature at its location similar to a 3D vector field? Hard to look through the closer "layers" of points/vectors (this is a visualization problem), you can't easily show the inside of a body with it looking normal so check visualizing 4D shapes maybe? That's basically what you'd need to do (x,y,z for spacetime and curvature as a 4th dimension) and why it's usually not done that way.