Although the audience doesn't find out Taiga has to transfer schools until Episode 24/Book 10, she found out about it when she visited her mother after the ski trip. Episodes 22-23/Book 9 show her acting based on that knowledge. Her refusal to complete the career assessment? It won't matter because she'll be off to another school next year, anyways. Her conviction that Ryuuji can still get Minori? Once Taiga moves, there'll be no reason for Minori not to go out with him. The chocolates she gives everyone? They were meant as farewell presents. Had Minori not interrupted, Taiga probably would've told everyone what was going on.
The only thing written on Taiga's letter-of-apology to Sumire was "Baka". Considering that she called her and her friends idiots, but later indirectly confessed of her affections to Yuusaku by lamenting that she also wants but cannot allow herself to become "an idiot who only knows how to dash forward" like them (as she knew Yuusaku would do anything to follow her all the way to America had he known), it was the highest compliment Taiga could give her.
The opening dialogue in the first episode, beyond its repetition in the finale, gets an easy-to-miss reference in the first of the Beach Episodes. The opening dialogue deals with Ryuuji and Taiga's thoughts on 'something that can't be seen', while Ryuuji and Minori's discussion in the Beach Episode touches on the issue of ghosts, which Minori describes as something she'd never seen. The latter came about as an answer to the question of whether Minori had ever loved anyone while it's pretty damned obvious what that 'something which couldn't been seen' is in the opening dialogue.
It’s a well-known axiom that infatuation is selfish, while genuine love is selfless, and the series uses this in a subtly brilliant way to show the viewer how Taiga's feelings are changing over time. In the beginning of the series when Taiga is completely infatuated with Kitamura, she puts more effort into obstructing Ryuuji's attempts to talk to Minori than actually helping him, even though they're supposed to be allies. When he complains about this at one point, she flat-out tells him that her happiness comes first and she won’t let him get near Minori until Kitamura is hers. Through the middle part of the series she still doesn't have Kitamura, but as her friendship with Ryuuji strengthens she not only stops obstructing him, she actually starts giving him chances to be alone with Minori and encouraging him to take some initiative. By the time the final arc rolls around Taiga has completely fallen in love with Ryuuji, but by that point she's made a complete 180 and is doing everything in her power to push Ryuuji and Minori together because she just wants them to be happy, even though it’s breaking her own heart to do it.
The reason that Ryuuji and Taiga decide against eloping towards the end. When Ryuuji discovers that Yasuko ran away before Ryuuji could run away from her, it appears that he and Taiga reevaluate their plan because they realize that they'd be doing the same thing that Yasuko had done when she was young. So rather than put her through that pain, they go to Ryuuji's grandfather and trick Yasuko into thinking Ryuuji had been hurt so she could see her parents and make amends for eloping and running away from them.
Here's a series whose main premise is the main characters realizing their love for their crushes has been shifted to love of each other. But it's all right there in the very beginning: the event that sets the action in motion is Taiga accidentally placing a love letter for Kitamura in Ryuuji's bag—literally redirecting her affections for Kitamura to Ryuuji, first without realizing, and then in spite of her resistance. Even more brilliant, the envelope for Kitamura is empty, much like her feelings for him are hollow: it's the reality of her life with Ryuuji that over time gives them substance.
The pink paper stuck on Ryuuji's house paper door shown later in the show is Taiga's love letter to Kitamura. She gave that to Ryuuji trying to help fixing Ryuji's door.
Why did Ryuuji keep so quiet after hearing Taiga's confession in episode 21? At the end of episode 4, when Ryuuji is reflecting on Taiga shooting down Kitamura's confession, Ryuuji opines that since Taiga isn't sure of her feelings for Kitamura initially, she "should've held off on giving him an answer". Ryuuji is practicing what he preaches.
Why does Taiga hit Ryuuji so often? She also hits her father often. Taiga has been living alone in an apartment and wasn't very good at taking care of herself. Then Ryuuji started to take care of her. He basically became something similar to a father figure for her, so this is why she treats him like she treats her actual dad.
Ami has blue hair and Minorin has red hair but everybody freaks out when Yusaku dyes his hair blond, and then they try to forcibly dye it black again. (Although, Blond dye is considered a hallmark of being a delinquent in Japanese culture, so there is at least some reasoning behind it.)
Ami (and the other blue-haired characters) can get away with it if you consider blue to represent a different shade of black. Minori, on the other hand, is a bit harder to justify. And then there's Haruta whose hair apparently is blue in-story, since he's the one Kitamura goes to to learn about hair-dying.
Minori can be just like Ami, a lighter coloured brown that's been stylised much like blue. Blue is a shade of black, Red is a shade of brown.
In the novels (Vol 6) it's stated Haruta dyed his hair blond over the holidays; since he's not blond any more, he must've dyed it back, so the blue is just the color his (presumably black) OTC hair dye reflects most.
