To access the companion quests you just need to get their approval rating high. Talk to them after every mission. Also talk to them more than once to make sure you get all the new stuff. Sometimes there is 2-3 things for them when I come back.
Flirting also helps with raising the approval rating if that is your thing. I only flirt with certain people based on how I want to play a certain character.
I don't try to do everything in one playthrough. I have 5 games going right now and each one is romancing a different person. Keeps the game varied and finding new dialogues based on race is always fun.
There are guides out there if you want all the details but I don't tend to use them as I want to see it first hand.