If the gun didn't come with it's box and/or instructions, you can download a "generic" S&W revolver manual off their website. It's a .PDF file located here
. It will provide you with some of the information you need to know.
Your gun appears to be a 5-shot .38 Special revolver with a 2-inch barrel. This would be built on what S&W calls their "J" Frame. The barrel should be marked ".38 S&W SPL" (as you indicated). With the exposed hammer and all-steel construction (as yours appears to be), it is referred to as the Model 36 - .38 Chiefs Special
. If the frame is non-magnetic (aluminum alloy) it is the Model 37 - .38 Chiefs Special Airweight
. Both are considered fine guns of excellent quality.
You say the revolver is "kinda old" - a relative term. It would not be uncommon for that gun to be in excellent condition even if it was made in the 1960's. Age can be determined if you provide the first half of the serial number, such as "123XXX" or "J 123XXX".
The .38 Special revolver is an excellent choice for self-defense and many members here own and carry them. For a first-time owner, you should know that the short barrel makes it more difficult to shoot with accuracy, especially as the distances get longer. But you can still learn to shoot your "snubby" accurately enough for self-defense with some coaching from an instructor or experienced shooter.
As a first time gun owner, you'll find shooting can be a rewarding and challenging experience. Plus a lot of fun!