We don't actually know when Celebrian was born - but we can make a few guesses. The first time she is mentioned is way back in 1300 S.A., living in Eregion with her parents, shortly before they moved to Lorien. Eregion was founded around 700 S.A. - and I think that marks a good candidate for Celebrian's birth.
We know that elves didn't have children in times of war and upheaval. What better time to start a family than when you've just founded a nice, safe kingdom? The turmoils that marked the end of the First Age are over. You have Lindon as an ally in the west, Moria to the east - you're safe.
And then there's her name. 'Celeb-' is silver, while 'Rian' apparently derives from 'Crown-gift'. There's an obvious reference there to her hair (if, indeed, she had silver hair), as well as to her parents' names - Celeborn, and Galadriel (and yes, that is the same element - 'Maiden crowned with a festive garland'). So... if there's two meanings to her name already, why not assume a third? A silver crown or circlet seems a perfect ornament for the Lord and Lady of Eregion (Galadriel and Celeborn ruled there before Celebrimbor did); could 'Celebrian' also mean 'A gift from the silver crown'? It's speculative, but I think a birthdate of around 700 S.A. is plausible.
Around 1350-1400 S.A., Galadriel and Celeborn moved to Lorinand, the later Lothlorien. Since we know Celebrian didn't die in Eregion four hundred years later, it seems likely that she moved with her parents - and thus watched from afar as her childhood home was destroyed by Sauron's orcs. I can't imagine that was a very pleasant experience.
And then... nearly two thousand years pass before the next major event: the War of the Last Alliance. We know that Celebrian met Elrond at some point (she married him, after all) - could this have been when, with him as Gil-Galad's herald, her as part of the forces of Amdir of Lorinand? We don't know what role she took - elves didn't forbid women from becoming fighters, but she could equally have been a healer - but given what happened to Eregion, we can be certain she did take part. Amdir's forces were caught in the Dead Marshes during the Battle of Dagorlad - should we picture Celebrian anxiously waiting for news of her parents (remember, Galadriel's mother named her Nerwen, 'Man-maid' - she was a fighter, too!), or should we picture her clawing her way out of the swamp, fending off orcs with a mud-caked sword? Either is possible.
Celebrian spent the next couple of millennia with her husband (they married in 109 T.A.). Like most elves, they had their children fairly quickly, and fairly close together - the twins in 130, Arwen in 241. This isn't to say that Rivendell was perfectly safe - it sat in the eastern marches of Rhudaur, the first province of Arnor to fall to the Witch-King. Angmar rose in 1300 T.A., and for the next six hundred years Rivendell would have been in a state of tension. They were allied with Arthedain, and with Lindon - but the road between was owned by the hill-men of Rhudaur, and a dangerous route indeed. Although further away, Lorien would have been a closer friend.
And then, nearly 2000 years after her own children had been born, Celebrian found herself as a foster-mother to Arahael, son of the chieftain of the Dunedain. In fact, this was a role she continued in for the next five hundred years, fostering (with her husband, of course) a full eight future chieftains. Did she know, with some inkling of her mother's foresight, that her daughter would be bound to a future member of the line of Elendil - one of the descendents of the little mortal boys she cared for, watched grow up, grow old, and die? I don't know. Perhaps. Certainly she would have seen in them the likeness of their forefathers - both the ones she helped raise, the ones she had seen from afar as rulers of Arthedain and Arnor (and remember, Valandil son of Isildur was also raised in Rivendell), and even all the way back to their first ancestor: Elros, her husband's mortal brother.
Celebrian's injury happened in 2509 of the Third Age, while she was travelling to visit her parents in Lorien. This is not a happy time in Middle-earth: Sauron had returned from the East some 50 years earlier as the Necromancer, prompting the formation of the White Council. The Ring had just been discovered by Deagol, and in 2480, Sauron had sent orcs to fill the caves of the Misty Mountains (and reoccupy Moria). We say that Celebrian was just going to visit her parents - but given the new danger of the journey, how likely is that? Of course, she'd made the trip hundreds of times before, but she wasn't stupid: she knew that the High Pass was dangerous.
Far more likely, then, is that she was travelling as part of a messenger party, from Elrond (of the White Council) to Galadriel (ditto). Yes, she likely went along in order to see her family - but the trip as a whole had a different purpose. Given the surrounding events, I think I even know what:
In 2509 - the same year Celebrian was captured - Gondor was under siege. Cirion the Steward called for help - specifically, from the Éothéod of the northern vale of Anduin. In 2510, Eorl rode south, won the Battle of the Field of Celebrant, and founded Rohan. I can't imagine Galadriel would have let Cirion's messengers pass by without remarking on it - and sending messages to the rest of the White Council. When Elrond wanted to reply, Celebrian asked to travel with the messengers - and so the story unfolded.