They are rivals from relatively early into Squirrelpaw's apprenticeship. Squirrelpaw antagonized him first because she thinks of him as the #1 up-and-coming warrior in ThunderClan, and thus a fitting rival for her, the great Squirrelpaw. Brambleclaw would not have normally taken the bait from an apprentice, but he idolizes Firestar, and he feels self conscious about his kin so brazenly challenging him (he wants her to think he's cool so bad, not realizing she already does). They frequently try to show-up one another. Every other cat in the clan is like "oooh they flirting."
Before they're really official (but are totally a couple already), they have their first big fight, about Hawkfrost. Squirrelpaw/flight is right in her purpose but she doesn't articulate it well. She initially has some trouble understanding his complicated relationship with his father, as compared to her healthy relationship with her own (who, again, is also the cat he idolizes). Brambleclaw is too closed off about the fact that he's dealing with a lot of Feelings, because he doesn't want her to see him looking "weak". It's messy and they're both frustrated because they both really like the other one still. Squirrelflight dates Ashfur for a bit, and she really does like and respect him, but he's just not the cat for her.
Eventually they talk things out. Squirrelflight notably does not gloat when her suspicions about Hawkfrost prove to be true, which is an early building block of trust in their relationship. They are over the moon about each other for a while. ThunderClan cats joke for a while that even a rabbit might be impressed by those two. They never stop challenging each other, though, and can frequently be seen competing over the most arbitrary thing. This includes the occasional argument, but they always make up enthusiastically.
When Squirrelflight shows up with kits Bramble is: ecstatic. He's full-on dad mode he's so excited. Things continue to be really happy until the reveal. Brambleclaw feels hurt because in his mind this means that she didn't trust him enough to share the secret. And he feels a misguided sense of loss that his kits are no longer "his". (How he ultimately works this out with each of the Three is a whole separate topic.) He initiates the breakup. Squirrelflight is hurt because she feels that he isn't giving enough consideration to her position. It very much mirrors the Hawkfrost situation in that way.
The break-up is bitter but quiet. They just sort of simmer. Brambleclaw makes it a point to not change how he makes patrols, putting her in charge of a normal amount, but he speaks to her only as required by his duties. She tries to make small-talk on a few occasions, but he quickly finds a reason to excuse himself each time ("Sorry, I need to plan tomorrow's patrols."), so she stops trying.
Bramblestar's Storm? I'm sorry you have the wrong number.
The great battle happens, Firestar dies. Brambleclaw names Squirrelflight his deputy. He has come to understand that his shock at learning that his kits weren't technically his caused him to lash out more than he might otherwise, and feels regretful, but he has a hard time broaching the topic with Squirrelflight. Meetings as leader and deputy start awkwardly strictly functional, but slowly it drifts into them hanging out, them going hunting together, eventually essentially dating again. ThunderClan buzzes with the gossip.
They get back together. Cats who were around remark to those who weren't how it seems like when they were first mates. That includes how the two of them will spend half a day trying to one-up each other on some point about who behaved most appropriately during the interaction at the WindClan border earlier. Alderkit and Sparkkit get SO MUCH ATTENTION when they are born
Squirrelflight's Hope? Never heard of her.
After the events of Broken Code things are very tense. Bramblestar is just... not up to it anymore. Squirrelflight basically has to lead even though she doesn't actually have the authority to and it stresses her out (especially after being acting leader in TBC didn't exactly go smoothly). They fight a lot, but they always nest together. Bramblestar has a hard time getting through the night a lot, and she wants to comfort him.
After Squirrelstar becomes leader and Brambleclaw has retired it's a breath of fresh air. With her officially able to handle clan matters they are happy as can be, and some of the younger cats might describe them as sickeningly sweet together. Brambleclaw dens with the elders during the day but he's in Squirrelstar's den each night. He's able to be the big spoon sometimes again :3
Idk I love them you can't make me not love them no matter how many super editions you write, Erin.