In a co-op game, players one and three experience Prologue A, while players two and four experience Prologue B. Each prologue yields a separate achievement or trophy when completed, so make sure you set yourself up to experience the other prologue on a subsequent play through.
Prologue B is an on-rails sequence in an Osprey VS. You don't need to do anything here, although you can shoot down the Trilid and attack the Raibees that appear. Your Machine Gun has infinite ammo for the time being, so go crazy.
In Prologue A, enter the vacant Nida VS in front of you and then press the X/Square button to activate it. Once in the VS, start moving northeast. This VS is equipped with dual Gatling Guns, which you can use to kill the Sepia akrid and destroy the Genessa pods you'll eventually encounter
Activate the Data Post to add 500 points to your team's Battle Gauge and reveal the map on the Radar in the top-right corner of the screen.
After the cutscene, activate the nearby Data Post by approaching it and rapidly tapping the B/Circle button. The benefits of activating a Data Post are manifold. For one, activating a Data Post adds 500 points to your team's Battle Gauge, which amounts to one extra respawn for you or one of your (human) teammates. Furthermore, activating a Data Post will reveal the map in the top-right corner of the screen, with nearby enemies appearing as red blips.
Once you've activated Data Post 1, approach the cliffside to the north. You must use your anchor to reach the top of this hill. To use the anchor, aim the target reticule at a ledge and press the X/Square button to fire the grapnel. The anchor will only latch on to something when the target reticule is green. If you're aiming at the ledge and the target reticule is white, try moving closer. When you've made it up, use your anchor again to reach the higher ledge.
Atop the hill is a T-ENG chest. To open these chests, you must equip your T-ENG Supplier and shoot enough T-ENG at the chest to open it. Equip the T-ENG Supplier by pressing Y/Triangle and LT/L2 at the same time. With the T-ENG Supplier equipped, shoot a couple of globs of T-ENG at the container to open it and take the Rocket Launcher inside. Once you have that, be sure to switch back to your grenades by pressing Y/Triangle and LT/L2 again.
Use the T-ENG Supplier to open T-ENG chests.
A swarm of Trilid will appear as you approach the other end of the hilltop. Gun down as many of the Trilid as you can with your Machine Gun and a couple of [?] Boxes may drop. These mystery boxes usually hold credits, but may also yield Emotes, Noms de Guerre, abilities or weapons. Credits can be used to play the LP2 Slot Machine, which you can access by pressing the Y/Triangle button while on one of the Customization screens in the 'My Page' section of the game's main menu. The LP2 Slot Machine is 2000 credits a go and will yield either an Emote, Nom de Guerre, ability or weapon each time.
Continue north to the other end of the hilltop and drop down to the area below. Activate Data Post 2 and then approach the nearby building. There is a mass of Sepia and several Genessa pods on the other side of this building. Use Hand Grenades or a Rocket Launcher to clear them out. There's another T-ENG chest here; open it to find a VS Shotgun.
You'll meet up with the other half of your team at this point, and then a large Chryatis akrid will emerge from the ground. One clear disadvantage akrid face is that their weak spots glow orange. The Chryatis's weak spot is its glowing orange abdomen. As well, you can shoot off each of the Chryatis's arms by blasting the mass of orange protrusions near each joint. Sever both of its arms and the Chryatis will fall to the ground, unable to attack.
The VS Shotgun will make short work of the Chryatis.
With the powerful VS Shotgun in your hands, the Chryatis will fall in no time. The Chryatis's attacks are damaging, however, so make use of the evasive roll manoeuvre by pressing the A/X button while crouching (click the Left Stick to crouch). Your character is invulnerable during the first part of the roll animation, allowing you to escape almost any enemy attack with the right timing.
When the Chryatis is dead it will release a bunch of [?] Boxes. These boxes disappear quite quickly, so get right to collecting and grab as many as you can. You can Dash by holding the B/Circle button.
After collecting all of the boxes, step into the red outline around the Osprey VS to complete the prologue.