Do any other waves work like light waves in that one "thing" gives rise to another thing? My understanding of light is that there are two types of fields involved, electro and magnetic fields and one causes the next which then causes the other type and so on.
Is light the only thing that works that way or does, for example, gravity work this way or when we speak of the wave form of particles, are the waves similar in this respect similar to light, maybe even electro and magnetic?
 A: 
My understanding of light is that there are two types of fields involved,
electro and magnetic fields and one causes the next
which then causes the other type and so on.
[...] are the waves similar in this respect similar to light?

You nailed it. This is actually the characteristic property of every type of waves.
Here are some examples.
One-dimensional waves in a chain of alternating springs and masses

(image from ResearchGate - Virtual musical instruments - Natural sound using physical models)
Here are also two types of fields involved: velocity of the masses
and compression/decompression of the springs.
The different lengths of neighboring springs causes
the velocity of the mass in between to change.
And the different velocities of neighboring masses causes
the length of the spring in between to change.
Sound waves in air

(image from ResearchGate - Macro, Micro, Nano)
Here again are two types of fields involved: velocity and pressure.
Spatial differences of pressure causes the velocity to change.
And spatial differences of velocity causes the pressure to change.
