Bastila is somewhat muted in her relationship with you before you leave Dantooine, but, like Juhani, will eventually open up and begin telling you of how she became a Jedi, after which you will get many and tedious lectures about the temptations of the Dark side, as well as ruminations on the after effects of being permanently separated from your family. If your PC is a man, then Bastila will become your love interest, if you treat her like a lady. Corellian ale, envirosuit walks on Manaan's ocean floor; the whole nine yards. Women eat that stuff up.
If you run around spaceports with Bastila long enough, you'll eventually run into a Twi'lek named Malare, who brings word of Bastila's mother, who is apparently gravely ill on Tatooine. Head to Tatooine, where you'll receive a tip that Helena is in the cantina; go talk to her, if you can stand the chill.
After speaking to Helena, you can pick up a quest that involves finding Bastila's father's body, somewhere in the desert, inside the krayt dragon lair that coincidentally contains Tatooine's Star Map. Once you find the holocron, bring it back to Helena in the cantina. You can freely give it to her, and then attempt to broker a peace between the ice princesses, or just let her know that you're keeping it, with the appropriate shifts in Force alignment.
Jolee has a rather Romeo and Juliet tale of love and loss to tease you along with after level-ups. He's an amusing old man, but harbors a deep reservoir of pain beneath his avuncular surface. Although Jolee appears in your journal, he has no real side-quest, like your other party members. He simply has an interesting story to tell about his life before Kashyyyk and how he came to land on that planet.
After your initial interrogations, HK will reveal that some of his primary memory functions have been obscured from him, perhaps from physical damage. As mentioned in his character bio, you will have the opportunity to Repair him and restore him to his full functioning, but, like Jolee, he has no real side-quest to speak of.
Mission's backstory revolves around her search for her brother, Griff. It will take some time after you first meet Mission to reach the terminus of her story, which comes when she continually insists that the search for Griff won't interfere with your mission. Once this occurs, take her into your party and start hitting the starports and towns. Eventually, the Lena from her tale will appear, and let you know that Griff has been captured by the Sand People on Tatooine.
You'll need to follow along the main plot on Tatooine, involving the Czerka's attempts to rid themselves of the Tuskans, in order to free Griff. You'll either be able to free him alongside the Jawas when you return with the moisture vaporators, or simply kill all of the Tuskans and set him loose then.
After Griff is free, he'll return to the Czerka offices, and ask you to retrieve a tach gland for him. These are located on Kashyyyk; you'll need to grab some from one of the bins near the poachers in the Upper Shadowlands and return it to Griff to complete the quest.