The character of Batman had previously debuted in the pages of Detective Comics in 1939 created by Bob Kane and Bill Finger. The following year, the popular character, appeared in, not only Detective Comics, but in his own title also.
Golden age comics were immensely dense and lengthy. The story length of a single issue rivals modern day multi-issue graphic novels. The Batman character had actually been a very dark and serious character until DC added his sidekick Robin to make it more kid friendly. Many golden age stories were silly, cheesy, and simple but Batman, early on, was far darker and more grim. After Robin came along the stories began to take a more simple tone. Robin had already debuted prior to Batman issue 1 and is featured on the cover.
The first feature contains the origin of Batman titled: "The Legend of the Batman - Who he is and how he came to be!". The original origin, which has been retold and tweaked many times since 1940, was told like this in Batman 1 -
Fifteen years earlier (which seems to put Batman in his mid-20's) Thomas and Martha Wayne are shot and killed in a robbery. There is no iconic "pearls on the ground" scene nor are they coming out of a theater. In fact, there is no context given, just a few panels of the robbery.
We then see a praying young Bruce Wayne vowing (presumably to God) to avenge their deaths. They then show, in one panel bursts, Bruce Wayne studying to become a master scientist (a skill no longer mentioned), lifting weights and training, and then studying as how to disguise himself. That is when a bat flies in the window and he finds the inspiration to become the Batman. This is all accomplished in one page.
We then go to the next feature which has no title but features a splash of The Joker. The Joker is hijacking the radio waves and broadcasting threats to kill specific people at an exact time. Of course, these people fall over dead at the exact time predicted and their face twists and contorts after death even while surrounded by police protection.
Everyone is baffled how The Joker is getting away with it and Bruce Wayne vows investigate. The Batman sneaks into the home of the next target of The Joker who is a rival mobster. A fight breaks out but ultimately The Joker shoots the gangster then tries to get away but Batman hangs onto the car. A fist fight ensues and The Joker actually gets the best of Batman and ends up throwing him off a bridge. Batman swims to safety and returns home to regroup.
The next victim is a Judge and the Joker has disguised himself as a police officer this time and murders him as well. This time Robin has taken it upon himself to stake out the Judge's home and ends up getting knocked out from behind by the disguised Joker.
Batman tracks Robin using a flashlight gadget and ends up in another fight with the Joker. This time he beats The Joker in hand-to-hand combat but The Joker uses a gas gun to escape. Batman, however, manages to rescue Robin this time and the two trail The Joker. The Joker's gas gun is now empty and the duo manage to capture The Joker and jail him.
The next feature has Batman in solo action against a giant brute. Bullets don't harm him and he is incredibly strong and is reeking havoc downtown. Batman trails the giant in an early concept Batplane. He ultimately finds multiple giants inside a hideout being controlled, apparently, by a stereotypical 1940's, bald, mad scientist.
The scientist injects Batman with the same serum that has created the giants and tells him it will begin to take effect at exactly noon the next day. Batman fights his way out only to end up face to face with multiple other giant monsters. Batman manages to fight off even these giants and ultimately the giants turn on each other and begin to fight.
Batman, being a master scientist, manages to concoct an antidote to the serum. He then jumps in the Batplane again and goes monster hunting. He hangs one of the monster with a rope and flies him until he's dead. Yes, Batman would occasionally kill or even use a gun early on. He finally hunts down and kills the last monster by knocking him, King Kong style, off a skyscraper.
Yet another feature, this time featuring both Batman and Robin. Robin has infiltrated a luxury cruise to prevent a jewel theft. It's essentially a Robin solo story until about three-fourths the way through when Batman arrives. This story introduces us to the character that would ultimately become Catwoman. Yes, the story also features a health dose of misogyny.
The Joker returns in yet another feature. The Joker has escaped jail and is killing people again. The Joker's body count in this issue alone must be about ten. Batman and The Joker again fist fight with The Joker again winning (The Joker is now best two out of three fights with Batman). Batman is just about to be unmasked when he springs to life and escapes.
The Joker hatches another jewel theft plot (these are very common in the golden age) which, of course, Batman and Robin again foil. There continues to be a lot of interaction between The Joker and Robin which would become a huge plot point by the 1980's.
At the end of the final fight it looks like The Joker has received a mortal stab wound and it appears he dies until the last panel when it is revealed that, somehow, The Joker survived. The issue then ends.