Jacob Lee is out on a job with his copilot Max. Max has some misgivings that Jacob is brushing off until an alarm in the cargo hold interrupts them. Jacob checks it out but things get one of hand really fast.
There is not a lot of gameplay in this chapter and so all players need to do is to head to the indicated objectives. Here is a guide to the first chapter of The Callisto Protocol.
Walkthrough (Chapter 1) – The Callisto Protocol

You start the game as Jacob Lee, the pilot of the UJC Charon cargo ship. His co-pilot max notices a yellow light in the cargo hold and Jacob goes to check it out.

You will now be in control of Jacob. Head down the hallway heading into the cargo hold, there will be an audio log in the room to your right if you want to listen to it.
In the cargo hold, there will be a few containers with radioactive symbols that Jacob will inspect during a cutscene. He will then say that he should check out the rest of the ship

From the containers, turn right into a long hallway filled with metal and pipes. Go into the door to your right and jump on the box to continue to the next room.

An alarm will be ringing and to you right will be a hallway on fire. There will be a breach and the hull and Jacob will inform Max saying they have some unwanted guests on their ship. You will now have to make your way back to the cockpit.
As you pass through the next room, you will see a couple of armed Outer Way terrorists through a small window. This next area will be a slow tight squeeze. You will see and hear more people as you pass through. These are all scripted and so you do not need to worry about stealth just yet.

You are almost at the cockpit, you just need to head up some stairs. Unfortunately, the terrorists are here too and you will have to sprint if you want to avoid them and their shots. Once inside, there will be a cutscene where Jacob opens the airlock to get rid of the terrorists.
The terrorist manage to break the window leading into the cockpit causing the doors to blow open and risk Jacob and Max being flung into space too.

You will now have to slowly crawl your way to your pilot chair. Keep moving and you will eventually have Jacob strapped in and try to save the ship.
Despite their best efforts however, their ship crashes into Callisto.

Jacob is seemingly unscathed but Max is not as lucky and dies shortly after the impact. Callisto security arrives and initially is here to help out.
After finding one of the Outer Way terrorists however, they arrest Jacob too and the chapter ends.