Welcome!
Any S&W .357 will have it's serial number on the bottom of the grip frame. You may have to remove the grips in order to see it. There maybe a letter off to the left of the number, it is part of the serial number.
If the gun is old enough, it will not have a model number. Model numbers are stamped on the frame in the hinge area you see when the cylinder is open. It will look like: Mod 19-3 or something similar. There may be other numbers in that area which are not used for identification purposes.
If you post some pictures I am sure someone will be able to help you identify it.
Again Welcome!