Finding your Smith and Wesson serial number depends on the type of firearm you have. Most modern Smith and Wesson revolvers have the serial number stamped on the butt of the gun's frame, in between the grips. Most of Smith and Wesson's automatic pistols have the number stamped on the side of the frame. Some of the company's other guns have the number in other locations.
Smith and Wesson's Sigma Series automatic pistols have both the serial and model numbers stamped on a small plate on the bottom side of the frame at the front of the gun. The SW99 automatic pistols have their serial numbers stamped onto the gun's frame at the rear, directly under the firing pin housing.
Some Smith and Wesson revolvers have a duplicate serial number stamped inside the yoke, along with the gun's model number. These numbers are only visible with the gun open. Revolvers manufactured before 1957 have nothing stamped inside of the yoke.
As of 2015, Smith and Wesson does not maintain any kind of database for finding manufacturing dates for their guns using serial numbers. However, you many be able to find out information about your firearm by contacting the company directly.
Keep in mind that Smith and Wesson only began using serial numbers in 1907, decades after the company began manufacturing firearms. Finding information about guns older than this typically requires expert assistance.