A lot of the elements that we think of for the character are in the serial. These include:
- Alfred the butler.
- Commissioner Gordon.
- The Batsignal
- The Batcave (complete with shadows of bats in the background)
- Using a grandfather clock to get to the Batcave
- Vicky Vale (a clone of Lois Lane)
- Batman's utility belt
I know that at least some elements like Vicky Vale were added to the comic book after originating in the serial.
Now, understand that the production values of the serial were low. Very few sets were built for it. Most, maybe all, were reused from other locations. The same is true for the location shots. Western were common and many of the locations look it.
The serial revolves around a masked villain called the Wizard. He steals a remote control machine that lets him control any vehicles within miles. It can also cause fires in cars. Typical for serials, the plot takes a number of twists. The machine burns out diamond bearings and needs replacements. The Wizard also tries to gain control of a new explosive which can also be used as a power source. The builder of the machine also built a machine to jam it but using the two machines together along with an ugly pendant will turn someone invisible.
A few observations:
Batman/Bruce Wayne drives a new Mercury convertible. When he is Wayne the top is down. When he is Batman the top is up. He and Robin hide their costumes in the roof when it is down.
Shock absorbers might not be known yet. Batman's car bounces a lot when he brakes.
Batman has the only modern car in the serial. Everyone else drives an older model.
Gothan City must be pretty small. Chases always lead to the country. There are only a few tall buildings.
Bruce Wayne is supposed to be rich but he lives in a large house, not a mansion. The inventor gets the mansion.
Batman doesn't try very hard to hide his identity. In addition to driving the same car as Bruce Wayne (Vicky notices) he and Robin pull into Wayne's driveway and runs into the house while in full costume.
The Batcave doesn't have much furniture. Batman and Robin keep their costumes in a filing cabinet in the middle drawer (under "C" for costume?).
The Wizard is able to control vehicles using his remote control machine but there is no sign of a steering wheel or other directional control. He seems to do everything with switches.
The Wizard and his men use some sort of viewer that can see through doors. No indication how this works.
Batman's utility belt has whatever he needs. If he needs a cutting torch then sure enough, he has a big one tucked into his belt. There is no indication where the tanks to feed the torch are. Under his cape?
Batman and Robin use some sort of long glass tube as a gas mask. This only appears on their utility belts when they are about to get gassed.
Batman and Robin are lousy fighters. They are regularly overpowered.