We actually don't see their first meeting until quite a way into the series. Because most of the story is told from Yukino's perspective, at the beginning we really only see how angry she is that Arima has 'taken her place', not how they met or when they first started talking to one another. We find out that Arima had been watching Miyazawa for quite some time and thought that she was perfect, everything a person ought to be. There was a case of mistaken identity where he thought that she had written him a love letter, but when that turned out to be false, Arima did gain one thing: he realized that he was in love with Miyazawa. Of course, when he told her about it, she immediately shot him down without thinking about the consequences of her actions. All Yukino was thinking was that she 'hated' Arima and that she didn't want him to find out about her secret. Arima, of course, who had never even confessed to a girl in his entire life thus far, was devastated. Yukino's sisters suggested that rather than 'hating' Arima, she probably only had a complex about him, but she didn't dwell on the thought for long.
It wasn't long after that that Miyazawa made the amazing blunder of totally blowing her cover. Arima dropped by to let her borrow a CD, but she not only answered the door in her 'other self' garb -- she also jump-kicked him in the stomach. As even Miyazawa could foresee, Arima immediately figured out that the girl he'd known and fallen in love with was nothing more than a facade made to deceive others. The catch? He didn't really care. He was still in love with Miyazawa, though realizing that she had a hidden side to her was the catalyst for his development for the rest of the series. It was after this that Arima himself let his real self show. He basically blackmailed Miyazawa into spending time with him -- she thought it was because he didn't love her anymore, but it was quite the opposite. Arima loved her so much that he would do anything to keep her at his side, even if it meant getting rid of his nice guy image. Miyazawa finally cracked under the pressure and ran away, but they reconciled when Arima admitted that he only wanted to talk to her and be with her.
Even though things looked like they were progressing smoothly, the would-be couple hit a rough spot for a while because Arima was having trouble coming to grips with his new self. He loved Miyazawa, but he hated that she was bringing his hidden self to the surface. Because of his dark past and less-than-amazing biological parents, he feared that deep down, he was only 'evil' just like his parents had been and that Miyazawa was forcing this 'evil' side to come out. However, Arima underestimated Miyazawa's kindness and she easily broke through all of his fears, saying that the two of them should be true to themselves and get through this together. She even went so far as to punch him in the face to demonstrate that she was aware of the 'real' Arima and that she liked him better than the Mr. Perfect that he had always tried to be.
Well, they had made it through their differences and gotten even closer, but there was just one last problem: they weren't a couple. True, Arima had confessed to her, but Yukino had turned him down at that point and so far hadn't said anything to indicate that she felt otherwise. Arima went ahead and confessed again, saying that he'd always loved her even after learning of her true self. However, Miyazawa, who was not used to her real self being out, was plagued by insecurities and could never figure out the right time to answer Arima. She finally decided that there was no way he still liked her (even though he said he did!) and that she would try her best to pretend like they were just friends, which led to a very painful exchange between the two that left Arima heart-broken once again. Finally, Miyazawa realized that even though she couldn't say her feelings out loud, she could show them in another way. She didn't want to hurt Arima with these misunderstandings anymore, so during a council meeting, she quietly took his hand and they were a couple from then on.