Yes, absolutely you can (and should) shoot .38 Special through your .357 Magnum, it's a less expensive way to train. And if you have someone (responsible adult) in your household that has trouble with the Magnum recoil, the .38 round is just the solution.
Be careful, there is a lot of B.S. info out there about rounds that can shoot through other guns. I started a thread a while back exactly on this topic, there is a lot of great information in that thread. If you can't find it, I may come back and post it here if you like.
Here is an example of some B.S. advice. This guy (in his tongue-in-cheek video; jump to 3:30) says you can shoot 9mm and .40 through a .357 SIG:
.357 Magnum (MAG) and .357 SIG are very different, not interchangeable.
You can also shoot .38 Short Colt. Why you would want to I have no idea, but that can be a fun little round if it's loaded with blackpowder.
The .38 S&W thing is interesting. Some that are made for the older guns have a bullet diameter of .360" and some have a diameter of .358" so they can be fired in .38/.357 revolvers.
The only thing to keep in mind is be sure and clean the cylinder thoroughly after shooting "short cases" (.38's) or you may have some issues the next time you want to shoot .357's through that gun. The cases are substantially longer, and if there is a bunch of powder residue in the chambers, the cartridges may not be able to seat properly.