The Brotherhood of Steel Essential Quest starts after completing Main Quest Reunion. If you still haven't found this quest then you must visit the Cambridge Police Station after leaving from Fort Hagen. Or Simply fast travel to Boston Airport and Talk to any Brotherhood Soldier. Let's Begin with our First Brotherhood of Steel Essential Quest:
So you decided to Stay loyal with Brotherhood of Steel then let's begin with the very Essential Quest Reveille where you need to Investigate the Brotherhood of Steel airship.
Objective
Walkthrough
As the Start of the Essential Quest, you need to first investigate the Brotherhood of Steel airship flying overhead. Then head to Cambridge Police Station directly, if you have any doubt regarding the location then you can talk to any Brotherhood Soldier at the Boston Airport and he will guide you with the Cambridge Police Station.
On your way, you will be attacked by Feral Ghouls and mostly near the Station Entrance so be ready to counter back. Thus Ending the First Essential Quest of the Brotherhood of Steel.
Now that you have successfully made your way inside the Cambridge Police Station, you should check the Military Frequency AF95 and later track Paladin Danse to investigate more.
Objective
Walkthrough
At the Boston, Airport listens to the Military Frequency AF95 carefully. If you are traveling from the south of the Corvega Assembly Plant or from Lexington and College Square vicinity then also you will be able to listen to this Military radio signal.
Once you get inside the Cambridge Police Station you will be attacked by few Feral Ghouls and then Go ahead and speak with Paladin Danse who is a human male NPC. Thus concluding this Mission and beginning of the next mission.
Note: You use your Power Armor to park in the garage.
With Help of Paladin Danse, you must fix the auxiliary power and Retrieve the Deep Range Transmitter while fighting against the Synth.
Objective
Walkthrough
Paladin Danse will ask you to follow him and help him to fix the Brotherhood of Steel comm array. First head towards the ArcJet Systems and get the Transmitter which is required to repair the Comm Array. On your way there will be few Synths waiting for you and after clearing them head straight towards the lab analysis room.
In the lab Analysis Room, you need to hack the terminal to retrieve the terminal password. Later Open the door the head upstairs to the second floor to fight with turrets or you can use the terminal just beside you to disable them. You can also collect the Tesla Science Magazine near these turrets. Now head towards the elevator and then to the engine core, but first you need to power up the elevator.
Once are at the core don't forget to get the Fusion Core and the Junk Jet from it then hack the facility terminal to start the generators. Here you have to take down few more Synths by the old fashioned way or just turn up the engines to cook them up. Use the terminal inside the elevator to take it Up and then continue fighting with the Synths until you reach the control room and grab the Deep Range Transmitter for the Synth corpse.
Once you have retrieved the Transmitter, you must Escape the area with Danse. Go downstairs, talk to Paladin and inform him about your progress. Hand him the Transmitter and you will be rewarded with Righteous Authority. Thus concluding the Third Essential quest for Brotherhood of Steel.
Now that you have gained the Brotherhood trust and the Righteous Authority to be a part of, it's time to Support the Brotherhood Recon Team in the Fourth Essential Quest of Brotherhood of Steel.
Objective
Walkthrough
Once you hand over the Deep Range Transmitter to Paladin Danse he will ask you to return to the Cambridge Police Station and Join the Brotherhood of Steel by standing and fighting for their cause. If you have completed the Act 1 in the Main Quest then you will be introduced as an Initiate else you are in Act 2 then you will be introduced a Knight.
Now that you are a part of the Brotherhood of Steel you need to support the Brotherhood Recon Team which includes Scribe Haylen and Knight Rhys for your future assignments(Non-essential Quest). With this, the fourth Essential quest comes to an end and new Quest Shadow of Steel will be added to your log.