Pre-Series
Fluttershy was born in the city of Cloudsdale. Born a weak flyer, she was found bullied by other pegasi by Rainbow Dash. she ended up knocked out of Cloudsdale on the beginning of the history-defying race, landing closer to the ground, and first meeting the animals down under Cloudsdale just before the Sonic Rainboom went off. This scared many of the animals and she went on to calm them all down, earning her Cutie Mark in the process of it and becoming the prophosized member of the Elements of Harmony.
Season 1
Much of her involvement at first is much of a side-character, her mainly giving her aid whenever called upon.
When they began seeing signs of Peridot's arrival, Fluttershy volunteered to stay with Rainbow Dash at the Galaxy Warp, just to be sure that nothing was there to worry anyone. She found later that Peridot had arrived, and she was the first to be found by her, almost getting shot by her before Rainbow intervened. Fluttershy, in the end, went off back home and having to keep quiet for the next few days as Rainbow went with Peridot. She felt worried over the situation until Rainbow Dash finally came back.
Season 2 + 3
She is presented mostly as a supporting character throughout Seasons 2 and 3, with her in aid to the other Gems throughout the events of both seasons without any prominent moment on her part.
Season 4
During one morning feeding her animals, she ended up finding Zappy not far outside. Worried for the animal, she decided to take him in and try to heal him, naming him such as well. It took her a while to figure out how to feed him, though when she left to get her friends, she came back to find him gone, freaking her out. She went out to try and find him, finding him off in Beach City, but then told that she had to give him up and send him back to Homeworld. It took some convincing, but she did eventually agree and left him near the Warp Pad to be taken back.
When the Thunderbird arrived into Beach City, Fluttershy and Star Quartz helped calm it down when it got into the barn. From it, she told the others that the Thunderbird was just hurt, attacking by visiting Homeworld soldiers, and it needed help. She and the others managed to heal the bird before letting it go.
On Connie II's call, she and the rest of the Mane Six arrive in Starlight Glimmer's town to get their new friend back. Unlike the others, Fluttershy didn't question much of the town's current state of equality, considering it their way of friendship and that they shouldn't insult them for it. She changed her mind though later when she got captured and taken of her Cutie Mark, and she couldn't do much to help until she got her cutie mark back.
When the group went to get Sapphire from White Diamond, she was placed as a offering for the Human Zoo.
Season 5
Cluster Arc
Much like the other ponies, she wasn't told of The Cluster straight away, but went off to help Steven and Amethyst with their monster hunting mission. During the first mission to the Beta Kindergarten, she was able to understand the Gem Tortoise, and found out that it wasn't a Gem beast at all, but a Light Creature.
While she did help out the others throughout the endeavors, she had her doubts during the event, as most of the time it was the monsters who found them, but here it was the other way around, making her start to feel bad about it. She got some reassurance however from the others that it was going to be fine.
After their involvement with Malachite, Fluttershy was the one who decided to stay with Star Quartz, Jasper, and Lapis Lazuli as the others went to catch up with the others.
Season 6
When the Pink Diamond case started, she and the other ponies began their investigation on the matter by Pink Diamond's palanquin.
When she began to handle a fundraiser for animals back in Ponyville, she found that the Ponytones had ended up losing Big Macintosh's voice due to him getting involved in a gobbling contest. After convincing, and a bit of Poison Joke, she was chosen to join the Ponytones and replace the role Big Mac had for the upcoming show. Right before the show could begin though, Fluttershy freaked out and didn't show up at first, too scared to face the music. She was eventually encouraged to go out, and the show went on without much trouble.
While not shown exactly how, she and the others ended up captured by White diamond, and she wasn't able to escape until the humans arrived.