Every companion quest, except for the ones for Zaalbar and HK-47, are received by talking to a messenger that appears in the docking area of each world. They most often appear when leaving the Ebon Hawk. Each companion has a required conversation you must have with them for their messenger to appear. This conversation is not available until after Taris, at the earliest. Also the required messengers will not appear until Bastila has rejoined your party. This is laid out in more detail for each companion quest section.
The conversation that is required for Mission's messenger to appear is the conversation about Lena. She'll talk about Lena as soon as you get on Dantooine if you wish. Once you have that conversation go to one of the dock areas on any of the planets except on Manaan or Tatooine and she should appear. After talking to her go to Tatooine and talk to the Protocol Officer about Griff. Persuade her to talk about him and Griff will appear in the Sand People Enclave where you can rescue him by completing the main quest far enough to the point that you are able to get the Chieftain's Gaffi Stick.
If you have made a peace with the Chief, you must ask for permission to let Griff go. After you have permission or have cleared out the compound, go ahead and talk to Griff and he'll go running off. If you took the brute force approach, you can actually kill Griff as long as Mission isn't in your party, for only a small amount of XP (though that's more XP than you'll see doing anything else for him).
If you do decide to rescue him, Griff won't appear in the Czerka Office until you see Griff's conversation with Mission. If Mission is in your party when you rescue him, they'll have their discussion then and there, but if not go to the Docking Bay, and it will happen automatically. For some reason Mission is unclothed in this conversation unless she is in your party. Remember that no matter what, don't give him a single credit. It doesn't earn you any LSP, XP or even a word of thanks.
Anyways, go to the Czerka Office and talk to Griff to get the second half of this side-quest. Once you talk to him about getting a Tach gland, you can talk to Mission to either persuade her to be angry towards Griff or to give him a second chance for DSP and LSP respectively.
Next stop: Kashyyyk. Continue on with the main quest until you have access to the Shadowlands and talk to Jolee. Go to the camp north of Jolee. There are two sources of Tach gland here. First you can either unlock or bash the middle metal box near the camp which contains 3 Tach glands. However the more profitable source are the Tach themselves. Talk to the commander and agree to work with him. This must be done before you run them off obviously. Killing Tach nets you no DSP so even light sided Jedi can go Tach hunting.
There are 17 Tach on this map spread about which you can kill for their Tach glands along with the three in the metal box means a total of 20 glands. Since you get 10 gold for each Tach gland, sell all but one of them to the Czerka Captain. Since you can't keep only one in your inventory I usually leave one Tach alive and kill it after selling the other 19 to the captain. Anyways no matter how you got the Tach glands once you have one in hand return to Tatooine and talk to Griff. He will be overjoyed and promise to pay you back for all you've done. Now no matter what he says don't give him a single credit, as you won't get any benefit from doing so, no extra XP, credits, LSP or DSP at all. Nothing good comes from giving him a single cent so don't do it!
Anyways leave the building and re-enter and Garm will talk to you saying that Griff has left. You won't gain any XP or LSP or DSP from this. Once this is done talk to Mission again (if you didn't piss her off against Griff earlier on that is) and you'll be presented with some more options to gain LSP or DSP. The real unfortunate thing is that besides said LSP or DSP there is very little benefit to saving his sorry self from the Sand People Enclave, so darksiders might find it more beneficial (not to mention fun) in slaughtering him then and there.
Bastila's messenger trigger is the conversation after she talks about her parents, the one where she says she has been measuring your progress. This conversation option is available after you gain your first star map. Malare will appear next time you trigger a messenger on any planet except for Tatooine and Korriban (you cannot have Bastila in your party on Korriban). If you're character is male and want to pursue a romance with her, it may be in your best interest in either finishing this quest as soon as possible, or not even getting the quest at all until after you finish conversing with her. You can avoid getting this quest by not having Bastila in your party when you are in the Dock areas of each world. However until you do meet with Malare, Helena won't appear in the Cantina.
The reason you might want to avoid this quest or finish it quickly is because once you begin this quest you cannot talk to her about your relationship until after you fulfill Helena's request. After the Leviathan you cannot talk to Bastila about your relationship with her at all (the reason will become apparent once you finish the Leviathan). Anyways once you get this quest go to Tatooine and Maana Demknot will fume about Helena (Bastila's Mother) in the Cantina. Make sure you have Bastila in your party when you talk to Helena otherwise she won't ask you to find Bastila's Father's Holocron.
Proceed with the Main Quest on Tatooine until you are able to enter the cave with the Star Map. Next to the Star Map there is a pile of rubble with the Holocron. Take the Holocron back to Helena where you have two choices on how to deal with the situation. You can either just leave before things get ugly (which will permanently terminate any possibility of future romance) OR you can help Bastila fix her relationship with her mother. No other dialog choices have any consequences so say whatever you wish. Either way you get 250 XP and keep the worthless holocron. And don't worry the credits that Bastila gives her mother doesn't come out of your pocket.
In any case once this side-quest is done you can continue to have further conversations with Bastila if you are male, which may end in Bastila falling in love with your PC but only if you helped Bastila reconcile with her mother. Also make sure that you don't talk to Bastila about your relationship after the "kiss" dialog otherwise you will set the romance variable to off. Your romance with her will affect a dialog that you will have with her later on (in the temple), but it does not change the ending in any way.
Canderous' messenger is first able to appear after talking about how much the Mandalorians lost in the war. This conversation can be accessed as soon as you have Bastila in your party again. Jagi confronts him about his actions in the battle of Althir and challenges him to a duel on Tatooine. Head over there with Canderous in your party, and meet him just southwest of the Sandcrawler on the tracks located there. This will trigger the appearance of Jagi with two Rodian thugs. While you can handle their conversation in two ways, you really want to persuade Jagi, no matter your alignment.
If you decide to just kill Jagi, you'll have to fight him and his two thugs, but they aren't too difficult. On the other hand, if you can talk out their problems, and make a difficult persuade check (either one), Jagi will admit his fault and commit suicide (you'll get XP for his death), and you get a bonus of 1800 XP on top! Then you can take on the two thugs afterwards. Since Canderous's dialog doesn't change either way and no DSP or LSP are rewarded for any actions, the only real difference is the XP reward and what the quest log says.
While Zaalbar has no messengers or any meaningful conversations, and his Quest is started automatically while proceeding forward with the main quest on Kashyyyk, whether you want it to or not. As soon as it starts, he becomes unavailable for the duration. While this side-quest does not need to be completed, Zaalbar will remain unavailable to the end of the game if you don't. For more information see Kashyyyk's side-quests.
Juhani's messenger, Xor is unique in that you need to trigger the messenger twice. After talking to Juhani about how her mother died and how she was sold into slavery earlier in life, Xor will approach you if you have Juhani in your party (not on Dantooine). This conversation has no special triggers required to make it available, however you will need to level a few times after talking to her for the first time before she spills the beans. Once Xor approaches you, no other messenger can appear ever again unless you download the bugfix here that I have, in which case no new messenger will appear until you finish this quest.
Xor will approach again later on and attack your party (again not on Dantooine). I have found that he most often appears when I transit back to the Ebon Hawk. Take him and his goons down, upon which Xor will have his last dialog before his death. At this point you can either persuade Juhani to strike him down, or to let him die on his own (after you have critically wounded him?) Anyways he ends up dying either way and you'll gain 1800 XP if just let him die or 1500 XP + whatever you gain from killing him if Juhani strikes him down as well as some DSP. Afterwards you can have a quick talk with her about this Xor incident, but afterwards it doesn't really affect your dialog with her at all. Too bad you can't turn her to the Dark Side or change her alignment in any way. She doesn't gain any DSP for the act which is not exactly fair or in keeping with the way Alignment works in the game.
Jolee is the only companion that has no required conversation for Davin Kotras (his messenger) to appear. Just leave Kashyyyk with Jolee in tow, and leave the ship with him in your party and Davin should appear to tell Jolee about how Sunry is accused of Murder. Go to the Ahto West in Manaan where you will meet Elora, Sunry's wife. It is best if you have Jolee in your party for this, as it will allow you to talk to Elora later on about the trial.
Anyways after talking to her about her husband, go west and south to the Ahto High Court. There talk to judge Shelkar and ask to be appointed as Sunry's arbiter for 100 XP. Now before you talk to the Warden in the Jail to start the trial it is in your best interest to collect as much information as you can as you gain XP for it. You can talk to any of them in any order that you wish, but usually I talk to Sunry last for reasons that will become obvious below. Here are the list of people you can talk to for evidence and XP.
If you decide to go through with the trial, your goal will be to get at least three of the judges up to +3 points to have Sunry declared innocent and you'll gain 400 XP for this. If you get all five judges to have at least +3 points, then you gain a total of 500 XP instead and the Sith are found guilty of interfering with justice and must subsidize the Republic's purchases for one year. If you fail to convince at least three judges of his innocence, then you only gain 300 XP and he is found guilty and must serve life in prison. The following list will show the dialog choices and how it affects each judge's opinions.
Carth's side-quest requires you to talk to him about his wife and son before the messenger shows up. Carth requires that you have at least two star maps before he will talk. Anyways once that's out of the way, on any planet but Korriban Jordo should show up afterwards and give you this quest, so long as you have not yet eliminated access to the Sith Acadamey. If you're character is female and want to pursue a romance with him, it may be in your best interest in either finishing this quest as soon as possible, or not even getting the quest at all until after you finish conversing with him. You can avoid getting this quest by not having Carth in your party when you are in the Dock areas of each world. However until you do meet with Jordo, Dustil won't appear in the Sith Academy. The reason you might want to avoid this quest or finish it quickly is because once you begin this quest you cannot talk to him about your relationship until after you finish saving Dustil.
After the Leviathan you cannot talk to Carth about anything exept for the default Dialog unless you have already have established a relationship with him. Anyways assuming that you do start this quest Dustil will show up in the Academy, south of Mekel. Unless you have Carth in your party you can't continue this quest, so I suggest that you have him with you. If you talk to Dustil you'll find out that he's more than happy being a Sith and isn't particularly happy about meeting his father again. If you say anything besides "Calm Down", Dustil will end up attacking you and dying. If it ends up like this then any possibility of a romance with Carth dies along with Dustil (not that it matters for male PC's).
On the other hand if you do tell him to calm down he'll give you a chance to offer proof that the Sith are evil. If you already have the proof Carth will present it automatically. You can find a hint as to where to find this proof if you hack into the computer in the Dueling room. Or you can just read where it is right here: in Master Uthar's room in a footlocker. The door to his room is locked but you should be able to bypass it with a rather low security check. While this is one of the few doors in the game that cannot be bashed open, if you follow the Double Cross side-quest Yuthura will give you a key to the room, so no need to worry if you don't have any ranks in security. Return to Dustil and Carth will automatically present the proof that you have, reconciling (at least in part) the rift between Carth and Dustil, and earning you 250 XP and a hefty amount of LSP.
While HK-47 has no actual quest or any messengers associated with him, but he does have a bit of humorous backstory as well as upgrades to his stats that you can get out of him. Any time after acquiring HK-47 you can talk to him to repair his memory. These repairs must take place before you board the Leviathan, after which you can no longer repair him. You can repair him up to four times, each successive time having a higher repair DC. The following table lists each repair, DC, and modification. The back-story with each repair is not included, to prevent spoilers. You always take 20 (automatically add 20 to your skill rank) when doing this check.
This means a minimum of 17 in repair is required to complete this side-quest. You can gain +3 from Calrissan's Utility belt, and another +3 from using master valor if you start with an intelligence score of 11 (10 if you have the Memory Package). This means that you need at least 11 points in Repair which can only be reached at level 8 for scouts and Consulars, otherwise you need 22 points to put in to this skill as it is cross class, and can only be reached at level 19! Is it worth investing the skill points? No, not really, as repair is not used very often, and the bonuses granted to HK-47 aren't all that great either. The only real good part about this quest are the funny stories (you really should try it at least once, preferably with a scout). Note that you cannot repair HK-47 any more once you finish the Leviathan, so you should finish this quest up before then. This is because this quests is forcibly ended when you leave the Leviathan, thus preventing any further repairs.