Here are some good tools that may help:

Forms Intersections First Aid Pack

and

eye.training - form intersections(this site has some other handy tools as well.)

I felt Antonio Stappaerts has a good breakdown of the process. Though it may be in one of his ArtWOD videos behind a paywall. He recommended trying to visualize the overlaps one cross-sectional slice at a time. Something like what I believe an MRI Machine produces.

Try to keep it simple at first, with not too many rotations.

There are also people on the discord that have made other helpful diagrams and breakdowns. betweenskyandsea is one person that comes to mind.

For an easier version to build up from, doing box intersections as a warmup could help.

But it could also help to change the way you approach the intersections.

Instead of thinking of the intersections as between a "box" and "sphere" you could instead think of them as between surfaces. What does this mean exactly? This diagram shows how the surface of the box and sphere interact. The box has flat surfaces which make the sphere have sharp curves in image 3, but when the box changes to a curved surface in image 4 the interaction with the sphere also changes.

Thinking of them in terms of surfaces means that the exercise becomes simpler. Instead of seeing it as complex forms interacting with each other you can see it as flat on flat surface interactions (box/box, box/cylinder(flat side)/cone(flat side) / flat on round/curved interaction (box/cylinder, box/sphere) / round on round (sphere/sphere, sphere/cylinder, clyinder/cylinder) interaction. This works for any form interaction not just the simple ones in this exercise. You could go through this album composed of multiple different interactions made by optimus on discord, and use this thinking to see how these surfaces interact. Notice the different interactions between the surfaces (not the forms) and you'll notice a pattern which may help you when you do form intersections.

If, after this, you still struggle you could also follow this guide to use paint 3D to make your own forms to see how they interact.

If you have any questions / if anything is unclear don't hesitate to reply.