Bran:

Sansa:

Arya:

Jon:

Grey Worm:

Sam:

Tyrion:

Brienne:

Drogon:

Ser Davos and Gendry:
Bronn:

Dissatisfied with the ending? It's fanfic time!
In addition to his initial title of Bran the Broken, Bran Stark began to be known by another nickname: the Quiet King. He rarely held public events, and when he held court to hear pleas from his subjects, his answers and judgements were often very brief and to the point.
While his reign was not completely devoid of war (there was a small Dothraki incursion from Essos, headed by a Khal who was inspired by Dany's crossing and funded by the Iron Bank after the new Master of Coin racked up some debt), all conflicts were resolved with minimal bloodshed. After all, it's tough to sneak up on an army that's serving an all-knowing, all-seeing king.
Bran's life and reign was so long, and lasted through so many winters, that he gained a third nickname in his later years: Bran the Ancient. By the time he abdicated the throne and ordered the election of a new ruler, nobody was quite sure how old he was, though maesters at the time estimated he was well past 100 years. In his final command, he had his Kingsguard take him to the Gods Eye, a lake surrounded in mist that was said to be cursed — but was also said to have an island at the center with an entire grove of weirwood trees. The last his guard saw of him, he was on a small boat, disappearing into the mist, though Northerners swear they would often see ravens with blue-white eyes perched on the walls of Winterfell for many years after.
Sansa, who became known as the "Wise Wolf," ruled as Queen in the North until her death many years later. During that time, she worked to reunite and rehabilitate the great houses of the North, which had been decimated by the invasion of the Night King. House Umber never recovered, and Last Hearth has since become a memorial — at Queen Sansa's command — with the intention that the horrors of the Long Night and the later return of the Night King are never forgotten.
Sansa rarely left the North during her reign, choosing instead to send ravens or summon Tyrion to Winterfell rather than visit King's Landing. It's likely that the sadness that seeing King's Landing brought her was the reason, though when asked she always responded that it was simply "too warm" in the South.
Though she did have children and continue the Stark lineage, Sansa respected the final request of Tyrion Lannister before his death and called for a new King or Queen in the North to be elected. A lady of House Mormont — a cousin of Lyanna Mormont, hero of the Battle of Ice and Fire — was chosen as the new queen, becoming the first ruler of the North that wasn't a Stark.
Arya sailed west, and the last anyone heard of her was from one final raven sent from her ship, reporting to Sansa: "Land has been spotted, all will be well."
Little is known of what happened to Jon Snow. He was to be named Commander of the Night's Watch, but insisted on leading an expedition north of the Wall to lead the Wildlings back to their homes first, saying, "I had always wanted to be a ranger anyway." He was expected back after six weeks, but after a full year went by, it seemed clear that the man of many names — Jon Snow, the White Wolf, Queenslayer, the Bastard King, Jon the Undying, and perhaps even the King Beyond the Wall — never planned to return.
Later stories among the Wildlings tell of a leader who established a new settlement in the ~true~ North and spent his later years venturing into the wilderness, communing with the Children of the Forest and recording their history.
Grey Worm — aka Torgo Nudho — sailed with the Unsullied to Naath with a vow to protect the pacifist Naathi from any further raids by slavers. The Unsullied ambushed the first incursion, leaving just enough of them alive to send them back with a message to never come to Naath again. The slavers returned with a mercenary army that the Unsullied defeated without losing a single soldier. The third time the slavers returned, it was with a pledge to never again take slaves from Naath, and an offer to open trade for goods instead of people.
The Unsullied kept guard until the last of them died out, but Grey Worm abdicated his command much earlier than that. Moved by the pacifist Naathi and the memory of Missandei, he vowed to never commit an act of violence again and changed his name to Arlinio; Valyrian for "one who changes." For the rest of his days, he could be found living near the beach, chartering transportation to and from the mainland. He also got pretty good at telling jokes.
Sam worked his way up at the Citadel, eventually becoming the head archmaester of the Order of Maesters, at which point he relaxed many of the Citadel's rules. Being the last of his House, Sam decided to enact a true democratic election for the new leader of what was once House Tarly...but the people all voted for him anyway, so he appointed the younger Sam as the new ruler of the House, with Gilly serving as warden.
Eventually, Sam wrote his own history, revising Archmaester Ebrose's A Song of Ice and Fire to make it a little less boring.
After the dust had settled in Westeros, Tyrion's first diplomatic trip was overseas to Essos, to meet with Daario in Meereen. He could have sent a raven to notify Daario of Dany's death, but he felt it would be better received in person. With Bran's permission, Tyrion stayed for a while and helped install and maintain a new government there. He felt he owed it to Dany for all his failings.
Tyrion maintained a friendship with Sansa, frequently visiting Winterfell despite hating the cold. But by the time he made it to the Wall, Jon had already gone. Tyrion "pissed off the edge of the world" in his honor.
Tyrion was offered Casterly Rock, but found the memory of his family too painful to be reminded of so frequently, and thus a new head of House Lannister was appointed. Tyrion died younger than most of his friends, due to cirrhosis of the liver. On his deathbed, he quipped, "I suppose I should have seen this coming."
Ser Brienne of Tarth used her position as head of the Kingsguard to lobby Bran to officially allow women to be knights. He obliged. Since then, countless women have served the militaries of the Great Houses, and several have served as part of the Kingsguard with direct tutelage from Brienne herself. Brienne never married, and neither did Podrick, but the two served with honor for many years and retired together as friends.
Anguished over the loss of his brothers and mother, Drogon fled far from Westeros and gave Daenerys a proper dragon burial in the ruins of Valyria. Nobody except Bran knew where he went to, and after a while even Bran lost sight of him. But still, some of the people of Essos swear they hear his screams at night, which are said to echo all the way to the Shadowlands.
After a stint as Master of Ships, Davos grew tired of the fast pace of life in King's Landing and tendered his resignation. He moved to Storm's End, where he served Gendry as a valued advisor for the rest of his days.
As for Gendry, he married and had children soon after becoming lord of Storm's End, but always kept scouts positioned up the west coast of Westeros, with the vague order to "watch for a lone ship." He also had a habit of studying people's faces very closely for the rest of his days.
Bronn managed to get all of the brothels reinstated.