Here's how romance works in Andromeda. Talk to the character whenever possible, and if you can use the Flirt option, do so.
...and that's pretty much it. Unlike other Bioware games, romantic interests are not influenced by whether you listen to their advice, buy them gifts, or commit actions they consider reprehensible. Flirt with them on ship, flirt with them during their loyalty missions, then flirt with them during the inevitable post-mission epilogue.
If this isn't working, it's one of two things-
1. You're not far enough along the main story. Some character missions come farther along the main timeline than others. In particular- while you can do Vetra's loyalty mission relatively early, her personal epilogue isn't available until just before the endgame. If your sweetie-to-be has locked you out, spend a little more time saving the galaxy. You might also check your email, in case conversation isn't triggering progress properly.
2. You're romancing someone else. While committing to a romantic relationship removes (almost) all other flirt opportunities, sometimes even significant progress along one track can block off others. This seems to happen most often with Reyes (progress with him blocks off others), and Vetra & Suvi (progress with others blocks them off). If you have one particular character in your sights, it's safest to forsake all others in pursuing them
Peebee and Liam can both be seduced early on, but a long term relationship requires the usual loyalty mission prerequisites (the former opportunity can be recognized by the character's stress- Peebee will be bouncing off the walls in frustration, and Liam will be doing angry, shirtless push-ups).
If you choose to just fool around with Peebee, rather than commit, romancing her does not initially affect other romances (she can be "discreet"). Likewise, choosing to wait with Peebee will grant a different romance scene later on.
Jaal's initial flirt options have a different symbol, suggesting friendship and camraderie. Keep selecting these to open up actual flirts later on.
Reyes is the only major romance that isn't a squadmate. Even occasional flirting will allow you to pursue a full relationship with him (you'll make a choice to commit during the party at Sloane's, and you'll have another choice when more about about Reyes' past is revealed)
Keri T'Vessa can be romanced independently of any other character (when you meet up at a bar, she'll even mention that she told the barkeep to watch out for your 'you know who.'). To win this asari's affections, flirt with her during every conversation- most of which will occur in the task Path of a Hero. After discovering Meridian, Keri will be free of her journalistic obligations, and you can meet her back on the Nexus to take things further.
Avela Kjar on Aya can be romanced by a male Pathfinder. She's pickier than most romantic interests in that you have to use flirt at every possible opportunity- miss even one, and her romance may not unlock. This means you'll want to be sure to talk to her on the Docks before rescuing the Moshae, and to complete both her questlines, Forgotten History and Recovering the Past.
Drack is the only squadmate that cannot be romanced.
Kallo & Lexi are the only members of the Tempest crew that cannot be romanced (though it's clear that Lexi has deep, abiding attraction to the Pathfinder that she's just too terrified to admit, even to herself).
For Scott Ryder: Cora Harper, Peebee, Gil Brodie, Reyes Vidal, Vetra Nyx, Avela Kjar, Keri T'Vessa.
For Sara Ryder: Peebee, Liam Kosta, Jaal Ama Darav, Reyes Vidal, Vetra Nyx, Suvi Anwar, Keri T'Vessa.