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For Auld Lang Syne - Recruiting Arcade Gannon
Arcade Gannon is at the Old Mormon Fort in Freeside. You can recruit him through a number of different means. If you're friends with the Followers of the Apocalypse you shouldn't have any trouble recruiting him. Completing "High Times" should also be enough. If you have 75 points in Speech you can convince him to come with you. The "Confirmed Bachelor" perk should do it too.
Note that he'll refuse to work you if you are liked by the legion. You should be able to get around this for recruiting purposes by just wearing NCR armor when you first meet him (and if you're liked by the Legion I assume you have some NCR kills under your belt).
Note that you'll need to be careful about your actions. Supporting the Legion tends to make him dislike you, and picking extreme options, like using Archimedes I to wipe out the NCR forces at Helios One, can be enough to make him leave you in disgust. You'll want to get through the quest quickly to avoid scaring him off.
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For Auld Lang Syne - Earning Arcade Gannon's Trust
Like most of the companions in New Vegas, you'll need to earn some trust with Arcade before he'll be willing to discuss his companion quest with you. Thankfully, this hidden trust system is really easy for Arcade. You just need two "points" in your favor and they're actually pretty easy to earn. Note that doing things that he doesn't like has no impact on his trust of you, unless you anger him enough to make him leave.
The absolute easiest path is to recruit him before you go to see Caesar for the first time. In this case, you can just drag him along with you. When you take the raft at Cottonwood Cove and arrive at the Fort for the first time, he should mention his objections to meeting Caesar. If you pick the option stating that you just want to hear what Caesar's up to and leave, you'll get two points. Note that this doesn't have to be true. It's actually the best way for a pro-Caesar/evil player to get the trust.
Another decent path, that should work for anyone, is to go to the Repconn HQ and walk around for a bit. Once Arcade talks about it, you should be able to earn some trust with him. If you find the crashed vertibird to the southwest of Camp Searchlight, you'll get another chance to earn some trust and complete the requirement.
If you bring him along and talk to Dr. Hildern at Camp McCarran, it should trigger a conversation. If you use the choice saying that you're glad that men like Arcade are still around, that should net you two points.
There are also opportunities during "The White Wash" and "Lucky Old Sun" if you choose to support Arcade or the Followers.
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For Auld Lang Syne - Triggering the Quest
Note that once you scrape together the two points of approval that you need, you will still have to advance the main quest to a certain point. Once you meet a trigger in the main quest, he should stop you (possibly right after you hit the trigger) and mention his troubled past and his plans.
NCR - After you kill House for the NCR and report this success, he'll be willing to bring it up.
House - One of the longest commitments, I believe he won't be willing to mention it until you take care of the Brotherhood of Steel for House.
Wild Card - After you kill House and install Yes Man, he should be willing to talk about it.
Caesar's Legion - There is no trigger in the Caesar's Legion quest line. The closest thing would probably be the ability to sell Arcade as a slave during the quest line. If you want to do this quest and recruit the remnants to fight for Caesar's Legion, you'll need to do the Wild Card quest. It's actually easy. After you kill House for Caesar, talk to Yes Man and have him install himself. You shouldn't suffer much of a reputation loss and you'll be able to do the quest.
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For Auld Lang Syne - Recruiting the Remnants
Once you gain Arcade's trust, the hardest part of the quest is actually over. The next part is a simple recruitment mission and you'll just need to travel around a little bit. Note that apparently the best way to guarantee proper fulfillment of the quest is by keeping Arcade with you at all times. After each recruitment, he'll have a small thing to say about each of the remnants. This apparently is a vital flag for marking them as recruited. Don't run away or fast travel until he talks to you about them.
Along that line, you'll note that there are two choices for each person's story. They basically boil down to telling Arcade that it's natural and good to remember one's glory days, or that they should forget their past and move on to forge something new. These are tallied for Arcade and impact his decision at the end (unless you have a lot of points in Speech, which makes it less important).
Daisy Whitman - You may have already met her in Novac. She has a room in the motel. It's the one at the end. Just go inside and talk to her at night, or find her in Novac during the day. She'll happily join and pass on her part of the passphrase for the bunker.
Orion Moreno - Another easy one to find. He lives in a marked house right next to the NCR Sharecropper Farm. Just go through the hole in the fence and you'll find it. Talk to him to recruit him.
Cannibal Johnson - It's only appropriate to hire Orion's enemy next. He lives in a small cave just to the east of the Aerotech Office Park. Walk over to it and up a curving path on the hill to find it. Be careful when you walk inside, because there's a bear trap by the door. You can find Johnson by the fire in the center.
Judah Kreger - A fairly easy one to find if you've been to Westside. He's easy to spot thanks to his bright red cap. Look for him resting against a wall inside Westside and talk to him.
Dr. Henry - This is the only complicated one. In order to recruit Dr. Henry, you will need to complete his quest "Guess Who I Saw Today." It's not a long quest, but you'll have to finish his research to convince him to join Arcade's cause.
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For Auld Lang Syne - Plans for Things to Come
After you get the last member of the group, you should have the Remnants' Bunker marked on your map. It's along the path to Jacobstown, although it's just a small hatch in a small cave. Walk to the marked spot and go down the hatch. If you recruited all five people, you should have all five words for the passphrase. Use the computer to enter the very stylish Enclave bunker.
Walk through the hanger and past the Vertibird to find the door to the briefing room. All five of them should be there.
Talk to Judah Kreger and he'll give you the big choice. You have to decide whether they'll fight for the NCR or Caesar's Legion. If you're supporting House or yourself and Yes Man, you will want to side with the NCR. It's the most compatible. Although it doesn't matter as much for the Wild Card path.
Choosing the NCR will anger Orion. He'll storm out to the hanger area. If you have 80 points in Speech, you should be able to convince him that the NCR are the better option and get him to agree to the plan. Reporting your success will net you Daisy Whitman's power armor and power armor training.
Note: Remnant Power Armor is all the same. It doesn't matter who's you get.
If not, he will probably attack you if you try to convince him. Gun him down at close range and take his power armor. Report the death to the other remnants and Daisy Whitman should be willing to give you the power armor training.
Choosing Caesar's Legion will disappoint Cannibal Johnson. He'll simply refuse to fight for them. There's no way to convince him otherwise, but he'll also leave peacefully. Talk to Orion to receive Johnson's armor and the power armor training to use it.
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For Auld Lang Syne - Arcade Gannon
If you sided with Caesar's Legion, this is actually easy. He'll call you a fool or a madman and storm off in anger at your betrayal. There's no way to convince him otherwise and that effectively ends his role in the story.
If you sided with the NCR, then things are a little more complicated. He'll finally voice his inner turmoil and explain that he's torn over supporting his father's legacy or keeping his own path with the Followers.
He can choose to join the remnants and wear his father's armor with them at the battle for Hoover Dam. Or he can go back to join with the Followers. From a material point of view, going back to the Followers is better, as this will give you his father's special Tesla Armor. It's your choice though and it obviously has major story impacts. Decide which story you like more.
Your responses during the recruitment part of the quest will determine his initial choice. He'll pick the side that you seemed to support more. If you have 80 points in Speech, it should be possible for you to talk him out of it and have him do the other, but if you can't pass the Speech check, you're out of options.
He'll take care of what needs to be done and then report back to you one last time.