Both Cloud and Tifa grew up in a countryside town, Nibelheim. Without much economic activity, the town came to almost entirely depend on Shinra after the company arrived to develop a Mako Reactor nearby. With Shinra--and stories of the elite military unit "Soldier"--increasingly permeating the news and the dreams of the youth, Tifa and Cloud grew up as any normal rural kids would. Tifa finding herself having to deal with the interest of a lot of the boys in town. And, Cloud, being an arrogant but lonely boy, putting himself above the other boys but also wanting to be friends with them. These scenes begin to be shown in Remake. This ordinary adolescent drama drastically changes when Tifa's mother dies, a traumatic event that would echo in both Tifa's and Cloud's understanding of their pasts and their connection to each other. The event is a catalyst in Cloud being shamed by many and eventually leaving Nibelheim, proclaiming to Tifa with boyish arrogance that he would become a war hero like the "Soldier" Sephiroth. He also makes a promise to Tifa that, due to the horrors he suffered, he sadly forgets.
Years later calamity hits Nibelheim. The town is burnt to the ground and most of its inhabitants die at the hands of Sephrioth, the very war hero idolized by the young Cloud years before. Surviving a life threatening injury in this event and with all members of her family dead, Tifa finds herself a young girl alone trying to survive in the slums of Midgar. The traumatic events at Nibelheim being hidden from the public by Shinra, that time only exists as memories in Tifa's mind. She has nothing and no one to confirm those memories with. That is, until she meets Cloud again in Midgar (shortly before the game begins). Beneath her outgoing and strong exterior is someone who is unsure about not only who she can trust but also her own past, her place in society and what she should do with her life. These layers become clear in Tifa's conversations with Cloud.
Backtracking a bit, while trying to survive practically alone in Midgar, she is befriended by members of the anti-Shira group "Avalanche". With few, if any, people she can trust, no friends and with her deep hatred of Shinra (the company that stole almost everything from her), becoming a member of the group comes natural to her. However, she becomes unsure about the increasingly extreme actions Avalanche is taking. Not too long after she becomes a member of Avalanche, she chances upon meeting Cloud in Midgar and the game begins. Things go on to spiral out of control, Tifa's fears manifest and she finds herself losing her second home, the Sector 7 slums. After this event, she crumbles when talking with Cloud, someone who has come to symbolize the only constant element in her life. Something that would cause Tifa pain but also solace as the story progresses.
Speaking of Cloud, even more than in the original game, it is clear in FFVIIR that something is off about him. More than any other character, he is who I referred to when I mentioned an unreliable narrator--and he's the game's protagonist! Parts of FFVIIR seem off and almost like a fever dream; this is in large part because Cloud's life is exactly that. Young Cloud went off to become a hero, but found himself becoming a nobody. After Nibelheim was reduced to rubble, in conjunction with Shinra's burying of the incident, Cloud was used as a test subject in Shinra's (particularly Hojo's) experiments, which included being injected with cells from Jenova, a superior life form in the eyes of Hojo, and being exposed directly to Mako. Cloud was eventually deemed a failure by Hojo and abandoned. Deeply poisoned, Cloud finds himself in Midgar and out of his mind. This is when he meets Tifa. Falling back on his personality when he was a boy, he puts up an arrogant and indifferent front, running from the failure that he thinks he is and his poisoned mind. There are numerous times in FFVII Remake when Cloud hallucinates that he is seeing Sephiroth and times when his mind blocks out what is right in front of him. There are also subtle moments in Remake of him attempting to break down his fake persona and figure out what is going on.
It takes a long time for Cloud to come to terms with who he is, what he has done and how to deal with his poisoned mind. The fact that Cloud is out of his mind doesn't help Tifa in her hopes to come to terms with her past and their shared past. Having questioned herself for so many years, she doesn't confront Cloud about his story and what seem to be lies until it becomes unbearable for her further into the story. From which point, their stories really begin to unravel and then be woven together again. This brief description hasn't gotten into what the deal is between Sephiroth and Cloud, Tifa and Aerith. Nor other aspects of Tifa's and Cloud's characters and the rest of the FFVII cast. But I'll stop here.
I find the story of negotiating the differences between memory and present--thoughts and reality--as well as the story of finding solidarity with others and of self-sacrifice to be meaningful and emotionally moving. Some might call that childish, but I would call it human. FFVII doesn't take itself seriously at times, which I imagine isn't palatable for some, but it has also made some gamers FF fans for life.
All of the characters' stories continue develop (including Barret's, since you mentioned him) in Rebirth and, judging by the original game, I imagine will further develop in the final entry to the trilogy. I have very high hopes and expectations for the game.