I warn you, this guide may contain spoilers!
Now this is where the fun begins. You have been granted the wonderful pleasure of having Arcade Gannon as your loyal companion. Simply talking to him via the companion menu will not do anything to unlock the quest. You have to gain his trust. And there are many things that you can do to do this. If you didn't know, in Fallout: NV there is a system where companions give you "trust points" whenever you do something they like. Once you have obtained enough trust points your companion will open up to you and ask you to do a quest for them. I believe every companion does this. So here is what you need to do to earn Arcade's delicious trust points.
There are a number of things you can do. I suggest you unlock the Explorer perk. This perk allows you to see everything on your Pip-boy's map, Including areas that you have not yet discovered. First, visit the Crashed Vertibird. Obviously Arcade has to come with you. Here is the location:
The vertibird is a pre-war aircraft that the Enclave used. There is a crashed one in FNV believe it or not. And no, you cannot fly it, but don't worry, at the vertibird you will find a Tesla Cannon Prototype! What is this TCP you ask? Well it is a shoulder mounted laser cannon! But it will be in piss poor condition when you find it. Anyways, back to topic. Go to the crash site. There you will find Pre-war Army robots guarding the crash site. You must defeat them. Advance toward the vertibird and gaze apon it's former glory. you have to wait there until Arcade talks to you. You will not earn the trust point unless he
talks to you.
Congratulations! You have earned a trust point! (That is if he talked to you about the vertibird.) But we are not done yet, far from it. We need more of his irresistible trust points. The next way to obtain a trust point is to talk to Thomas Hildern in Camp McCarran. He is inside the main building on the first floor. The enterance to his lab is near the staircase.
Talk to Hildern about anything. He will give you the option to do a quest about Vault 22. If you are in a hurry and want to get this done then do not accept his quest and come back for that later. Almost immediately after talking to Hildern Arcade will come up to you and talk to you. Say the following to Arcade:"Good thing there are still people like you around."
Saying this to him is what gets you the points.ONWARD! OUR TREK IS NOT OVER YET!
Next you need to visit the REPCONN Headquarters building. Once again, you don't get the trust points until Arcade comments about REPCONN. And thats it for that... Moving on!
Our next destination is... THE FORT! Dun dun dunnnn! *Thunder*
Well, most likely you are vilified by the Legion. So how do we get in? There are a few ways. The first is to kill a Legionary and put on his uniform, but Legionary mongrels can sniff you out and alert everyone. Your next option is the sneaky way. Good luck with that. Do you guys remember our old friend Vulpes Inculta? No, anyone? Hes the Legionary who wears the wolf pelt and talks to you in Nipton. He shows up a second time after you have first confronted Benny and when you leave the Tops Casino. He will meet you right when you exit the casino and give you the Mark of Caesar which allows you to enter Legionary territory without being attacked.
Now to actually get into The Fort you need to go to Cottonwood Cove. Once there you must go to the docks and get on the raft. The raft will take you to The Fort. And yes, the place is filled with Legionaries.
When you arrive at the Fort Arcade will kindly
ask you why you are at the notorious Fortification Hill. Tell him that you are there to see what Caesar is up to, after that you can leave.
By now Arcade should have given you the quest. If not there are a few more things you can do. If you have not done so already, complete the quest "That Lucky Old Sun" with Arcade by your side. When you are given the choice on where to send Helios One's power Arcade will suggest sending the power to Fremont and Westside. You can do two things here to gain points. You can agree, or you can divide the power equally. If you choose to divide the power equally you are going to need 7 Intelligence to say:"Redundancy. If Caesar takes the dam and cuts off power, it will be chaos all over New Vegas."The White Wash
To start this quest you need to talk to Lieutenant Boyd about missing people at Camp McCarren. She can be found inside the main building. You get points by keeping Anderson's secret and telling the NCR that Cpl. White was killed by Scorpion gang members.
You can also gain points by helping the FoA while having Arcade as a companion.