Tim, first, welcome to THR.
Yes, the butt, or bottom of the grip is the correct location for the serial number. The serial number you've posted tells us the revolver was made approx 1971. The K (or 3K in this case) prefix tells us this should be a target frame, meaning it should have an adjustable rear sight. Being chambered in .38 Special narrows it to the models 14 and 15. The 14 is the "Target Masterpiece" and will have a flat front sight that is squared at the back, and rounds quickly toward the muzzle (the front of the barrel). The 15 is the "Combat Masterpiece" and will have a ramped front sight. If you open the cylinder and look under the crane (the crane is the arm that holds the cylinder as it swings out) the model should be stamped in the frame under the crane. This should be a model 14-3, or 15-3 based on the SN. You may also see other numbers stamped under the model number: These may be factory assembly numbers used to keep track of parts while the revolver was being built, or it may be a repeat of the serial number. There may be no numbers but the model number as well. The silver coloring you note is typically nickel plating, and would have been a fairly common factory option for the time period. Let us know if you need more info.