Instructions
Talk to Ashley, if Shepard is male, or Kaidan, if Shepard is female, after the Eden Prime mission, when Shepard wakes up in the medical bay. If you pick the positive conversation options, this will initiate the romance option with that character.
Initiate conversations with Liara T'soni when she joins the crew. This option is available to either gender, as T'soni is bisexual. During the second conversation you have with her, she will ask whether you find her attractive. This initiates the romance option for her, although if you are already pursuing one of your human teammates, you'll have to choose one over the other.
Take the side of your partner during the assault on Virmire. At the end of this mission, you must choose whether to help or sacrifice one of your teammates. Choosing to help your partner strengthens this relationship; sacrificing them kills them. In the subsequent scene, arguing before the council, choose your partner's side; this should bring your partner's loyalty to you as high as possible.
Take the final mission to Ilos. On the Normandy, you'll get a scene in your private quarters with your love interest. If you choose the Paragon speech options, you'll get the romance scene. If you choose Renegade, you'll send the love interest away.