The Twin Lamps is an abolitionist society. While illegal elsewhere in the Empire, slavery is very legal and widely practiced in Vvardenfell.
The Twin Lamps is a faction of sorts, but you don't really join it in the traditional sense. There is no person to whom you can go and say, "I want to be a member." The way to join is to simply be an abolitionist. In other words, free the slaves. In this faction, actions speak much louder than words.
It's not too difficult to find slaves, but it can be difficult to find slaves that can be freed. Every slave wears a slave bracer, so finding the slaves is pretty easy. If you see a shiny bracer, the person is a slave. The hitch to the whole equation is that the freeing most slaves requires finding the key to their bracer and not all bracers have keys.
Bandit caves are pretty straightforward. If you find slaves in them, there is a key around somewhere. It may be laying around on a chest or crate, or it might be in a bandit's inventory. In some cases, there are multiple keys floating around.
Slave bracers in mines usually have keys, though the location is usually somewhere outside of the mine.
Slaves in slave markets have bracers with no keys. You must buy a slave and then set him or her free. Not all slave markets are willing to sell slaves, though.
"Cattle" are slaves in vampire lairs. I have yet to find a key for their bracers and most do not have a "go free" dialogue option, anyway.
Finally, slaves in wizard towers are pretty much stuck. No key.
If you ask a slave about "go free" and do not get a response that says that you must have the key, the odds are really good that there is no key for that bracer. If you do get such a response, you'll need to go hunt for it.
When you free a slave, close the dialogue and immediately open it again and ask about "go free". If you have freed 5 slaves, the sixth (and all subsequent) will ask if you belong to the Twin Lamps. You can follow up on the "Twin Lamps" thread to find out that it is an abolitionist society.
When you have freed 15 slaves, asking the the sixteenth (and all subsequent) about "Twin Lamps" will get you the password for the faction ("they light the way to freedom"). Once you have the password, you are a "member" and can start looking for quests.
I have found this to be a little hinky. The game does keep track of how many slaves you have freed, but you have to reopen the dialogue with each freed slave to make sure that you trigger the dialogue flags. This means two "go free" conversations with each slave. The first is to unlock the bracer and the second is to find out about the Twin Lamps.
No one really points you to Ilmeni Dren, the quasi head of the Twin Lamps. You just have to stumble onto it. If you are a member of House Hlaalu, you can pick up a couple of faction quests from her, and one of the Bal Molagmer quests from Jim Stacey sends you to her, but the Twin Lamps is something that you really have to luck into or be told about in a walkthru.
Once you have freed more than 30 slaves, Jobasha (Jobasha's Rare Books in the Foreign Quarter of Vivec) will give you three skill-raising books from his personal stash. Jobasha will also give you the Twin Lamps password if you have freed more than fifteen slaves and his disposition is at least 70.
Your first mission from Ms. Dren is to help another member of the Twin Lamps in Ald Velothi. It's more or less north of Gnisis and southwest of Khuul. If you are a member of House Redoran, one of your early Redoran quests sends you up there, so it's already marked on your map. One of the "special duties" for the Morag Tong will send you to the area, as will one of the Temple quests.
You need to talk to Galyn Arvel on the dock to find out that there is an escaped slave named J'Saddha hiding in the nearby daedric ruins and that there are three slavehunters combing the area looking for him. Your job is to escort J'Saddha to Arvel.
You have three methods available to deal with the slave hunters. The first is to simply try to escort J'Saddha to Arvel and hope that you aren't caught. This is, by far, the more difficult method. But it can be done if you are very, very careful and very, very lucky.
The second method is to simply take out the slave hunters. They will not attack you on their own, so you'll either have to provoke an attack or resort to outright killing them.
The third method is to get their dispositions up to the point where they will believe you if you tell them that he left. This takes at least a 60 disposition, but higher would be safer. If you talk to them individually and convince them that J'Saddha left, they will all congregate behind a hill on the south side of town and you can "sneak" J'Saddha from the ruins to the boat (half the town sees you do it, but it's important that the slave hunters don't see it).
Report back to Ms. Dren for your pat on the back.
Time to rescue another slave. This one is at the Dren Plantation (almost directly north of Vivec and east of Pelagiad) and requires a bit more ingenuity than the last. Hides-His-Foot needs to be freed and escorted to Sterdecan's Farm. Unlike J'Saddha, you aren't going to be able to simply misdirect the guards and sneak him out (a targeted "invisibility" spell, maybe?). Also, as soon as you free him or any other slave, the guards know about it and immediately go into attack mode.
This is where you need to exercise a few preemptive strikes. Try to take the guards out one at a time (taunt them into attacking you). Once that's done, you can just waltz right out of the plantation. But be warned that they are pretty decent fighters.
Sterdecan's Farm is northeast of the Dren Plantation. Just follow the road north and east past the Arvel Plantation and you'll see Sterdecan's shack on the left side of the road. Sterdecan is also the only Redguard in the area, so he shouldn't be too hard to spot. Once Hides-His-Foot gets to the door of the shack you'll get the "your journal has been updated" message and a "thank you" from Hides-His-Foot.
As an added bonus, he will tell you about the Ienith brothers in the basement of Dren's villa and about the note about assassinating the Duke. So, that makes another way to get that information.
Report back to Ms. Dren for your pat on the back, a boost to your reputation and a boost to Duke Dren's disposition.
Ilmeni Dren has no more Twin Lamps quests. Galyn Arvel can help you out with one of your Redoran stronghold quests. Jobasha will never admit to being a member and I haven't run into anyone else who has "slavery" or "Twin Lamps" as a dialogue thread.