An Omega serial number will either be laser etched in tiny digits on the case back or for older watches it may be on the inside of the case or on the movement. It will be a number (no letters) between 5 and 9 digits long. The number containing dots, spaces and possibly letters is not the serial number, that is the Omega case reference or model. These can be decoded using the Omega model decoder.
This tool can date vintage Omegas from 1895 to 2008. Speedmaster serial numbers did not run inline with non-speedmaster Omega serial numbers so please select which you have, if you have a speedmaster it will be clearly marked on the case back.