We Do Not Know
...when the winter started but we have a possible range and we know the year it ends.
We know from this wiki page that the series begins in 298 AC. From this we can work out the start of the summer and so the end of the winter.
"You are a young man, Tyrion," Mormont said. "How many winters have you seen?"
He shrugged. "Eight, nine. I misremember."
"And all of them short."
"As you say, my lord." He had been born in the dead of winter, a terrible cruel one that the maesters said had lasted near three years, but Tyrion's earliest memories were of spring.
"When I was a boy, it was said that a long summer always meant a long winter to come. This summer has lasted nine years, Tyrion, and a tenth will soon be upon us. Think on that."
A Game of Thrones, Tyrion III
The old men called this weather spirit summer, and said it meant the season was giving up its ghosts at last. After this the cold would come, they warned, and a long summer always meant a long winter. This summer had lasted ten years. Jon had been a babe in arms when it began.
A Game of Thrones, Jon VII
So from this information it puts the end of the winter around 288/289 AC.
We know it was summer when Daenerys was born.
She had been born on Dragonstone nine moons after their flight, while a raging summer storm threatened to rip the island fastness apart. They said that storm was terrible. The Targaryen fleet was smashed while it lay at anchor, and huge stone blocks were ripped from the parapets and sent hurtling into the wild waters of the narrow sea. Her mother had died birthing her, and for that her brother Viserys had never forgiven her.
A Game of Thrones, Daenerys I
We also know that Dany was 14 in the year 299 AC and so was born in 284 AC. See the following page for more information on this:
When Dany is following the comet, she mentions she is still 14 years old. The comet has appeared in the sky in 299 AC. This means that Daenerys has been mentioned to have been 14 years old in both 298 AC and in 299 AC, eliminating 283 AC as Daenerys's possible birth year, leaving only the option 284 AC.
Years after Aegon's Conquest/Calculations Ages
Looking at the following wiki page there's no more information on the end of winter/start of summer. As such we can say it must have started between 284 AC and 288 AC, probably from 285 AC to 288 AC though that is just a gut feeling of mine.
As for if anything happened in these years well:
- 285 AC: Cersei Lannister falls pregnant with Joffery Baratheon.
- 286 AC: No known important events happen.
- 287 AC: No known important events happen.
- 288 AC: No known important events happen.