This element of the books came later in the series. It started with the awkwardness between Ron and Hermione, and then the almost-relationship between Harry and Cho. They all strayed from each other at one point or another, but I think Rowling did a perfect job of bringing the right people together in the end.
Harry and Ginny: I imagine that Ginny fell in love with Harry completely when he rescued her from The Chamber of Secrets. In all the books (and movies), she acted like he was a god. But I think she realized eventually that Harry had his flaws and that he could feel something for her. Harry, on the other hand, seemed to stumble upon his love for Ginny. The first girl that he became infatuated with was Cho Chung in The Goblet of Fire and through The Order of the Phoenix. Alas, it was never meant to be. I can't exactly pinpoint when Harry began falling for Ginny. I think they're cute together, but I guess it was just unexpected. Nothing like Ron and Hermione's relationship.
Ron and Hermione: This relationship was a long time coming. They both had their little flings (Hermione in The Goblet of Fire with Krum, Ron in The Half-Blood Prince with that girl that called him her Won-Won and who's name I've blocked out of my memory because she was so annoying). But they'd always been drawn to each other, and no matter how much the other messed up, they got past it. We didn't get much of a chance to see Ginny and Harry together, but I think Ron and Hermione were perfect together. They make each other better people, and their dance around each other for the entire series added an amazing layer to an already complicated story.
Nevil and Luna
(they didn't end up together, but in the movie Nevil makes a great case about loving her)
Lupin and Tonks
Arthur and Molly Weasley
Bill and Fleur