How do you find an old trumpet serial number?

The brand name is usually printed on the bell of each Trumpet.

If you are trying to identify the key of your trumpet, most trumpets that students start with are in Bb. It is much harder (and a little more expensive) to get a C, D, Eb or Piccolo trumpet, etc. (anything other than Bb)

If you are a student starting out or someone who inherited the trumpet from someone else, there's about a 80-90% chance it's a Bb trumpet. If you are still curious, I'm sure someone who works at a local band instrument sales/repair shop would have no problem helping you identify what you have.