In KOTOR, most of NPC quests are triggered if you talk to your members enough times. Once you fulfill the condition, you may travel to different planet with NPC. Or you may stay on the same planet, exit the landing pad & transit back with NPC.
In order to fulfill romance with Carth & Bastila, you must complete their quest (in Carth's case, you must save Dustil).
Zaalbar's quest is linear with the plot, so I won't list here. But you will have two choices, side with Freyyr (Light Side) or Chuundar (Dark Side).
1) Mission: Mission's Brother
[Condition] Talk to Mission until she mentions about her brother Griff and his girlfriend Lena.
[People to meet] Meet Lena, then talk to Czerka Officer on Tatooine. Griff will be in Sand People's Enclave.
[Outcome] Depending on how you treated Griff & Mission, you may receive LS or DS points.
Light Side: Part 1 Part 2 Part 3 Part 4 Part 5 Part 6 Part 7 Part 8 Part 9
Dark Side: Part 1 Part 2 Part 3 Part 4 Part 5 Part 6 Part 7 Part 8
2) Bastila: Bastila's Mother
[Condition] Talk to Bastila until she mentions about her parents and how she didn't get on with her mother.
[People to meet] Meet a Twi'lek woman who tells you about Helena's illness. You'll also meet a Duros who complains about Helena. You can find Helena in Cantina on Tatooine.
[Outcome] This quest is neutral.
Part 1 Part 2 Part 3 Part 4
3) Canderous: Jagi's Challenge
[Condition] Talk to Canderous until he talks about Revan twice.
[People to meet] Meet Jagi & accept his challenge on Dune Sea on Tatooine.
[Outcome] You may kill Jagi, or he may take his own life.
Kill Jagi: Part 1 Part 2
Jagi takes his own life: Part 1 Part 2
4) Jolee: Sunry Murder Trial
[Condition] This quest is automatically triggered if you enter Ahto West on Manaan with Jolee.
[People to meet] Meet Elora, then apply for an arbiter in High Court.
[Outcome] You'll decide whether Sunry is guilty or innocent.
Sunry Guilty: Part 1 Part 2 Part 3 Part 4 Part 5
Sunry Innocent: Part 1 Part 2 Part 3 Part 4 Part 5 Part 6
5) Carth: Finding Dustil
[Condition] Talk to Carth until he mentions about his wife and son Dustil.
[People to meet] Meet Jordo and learn that Dustil has joined the Sith Academy. You'll find him there.
[Outcome] You can save Dustil from the Sith or kill him.
Save Dustil: Part 1 Part 2 Part 3 Part 4 Part 5
Kill Dustil: Part 1 Part 2 Part 3
6) Juhani: Threat from Xor
[Condition] Talk to Juhani until she talks about how her father was killed on Taris.
[People to meet] Meet Xor, then he will appear again in the private hanger on Manaan or Korriban.
[Outcome] Juhani will stay on the Light Side, or submit to the Dark Side.
Light Side: Part 1 Part 2
Dark Side: Part 1 Part 2