Here is how i think the TV series will play out:
Jon goes south. Finds out by raven he is a Targaryen after Bran spills the beans to Sansa. Jon meets with Dany, they get married and bang, because marrying within their own blood is what Targaryen's do. Dany teaches Jon to ride one of the dragons, and he is particularly good at it because he is both a Targaryen and a warg. They use the dragons to destroy the iron fleet and Casterly Rock. Theon saves Yara and kills Uron. Destroying her hometown and her fleet throws Cersei into a rage and she attacks the Dornish who are besieging Kings Landing, only to be defeated as Jon/Dany/both ride in on their Dragons and melt Sir Gregor. Cersei dies - potentially killed by Arya Stark. Qyburn also dies in some horrific fashion. One of the dragons will be wounded during the fight with the Lannisters (by the arbalest shown in the last episode). The 7 kingdoms then unite behind Dany. Sansa is named warden of the North. Petyr Balish tries to force himself on her and is rebuffed. He flees and is ultimately killed in some satisfactory way.
Jon and Dany bang a lot and begin mining dragonstone for dragonglass per Sam's advice. They also start collecting valerian weapons and sending them north. Meanwhile, the night king sacks east watch by the sea and either bypasses the wall or somehow causes it to fall. He then heads south with a massive and growing undead army. An army formed of the knights watch, the wildlings, and the brotherhood without banners attacks the undead army to slow them down and is defeated. Sandor Clegane survives and flees to tell Dany and Jon of the encroaching undead. Meanwhile, the night king and his army destroy town after town, with the new corpses being converted to wights.
Desperate, Dany and Jon lead an army north to fight - this time with Bran riding or warging the third (previously wounded) dragon (because the greenseer tells him he will never walk, but he will learn to fly). During the ensuing battle(s), the wounded dragon is killed and is converted to an undead ice dragon. Bran is saved but is gravely injured. While recovering/knocked out he has a green vision that shows him how to permanently defeat the knight king. He recovers enough to tell Jon and Dany, and then he dies. The show climaxes with a giant battle where Jon/Dany defeat the undead ice dragon and roast the undead army. The ice king is ultimately defeated by Jon or Dany (whoever turns out to be the prince who was promised), potentially by removing a dragonglass shard from his torso. Many big characters will die - Jaime, Brienne, Tormund, Dolorous Ed, Yara, Theon (redeemed), Davos, Lyanna Mormont, Benjen Stark, Greyworm, Sandor Clegane, Podrick Payne, Ser Bronne, etc. Speaking of Bronne - I think he will be reunited with Tyrion and ultimately brought back into his service.
In the end - Westeros will be bloodied but will survive. All will be hunky dory with Queen Dany and King Jon. Final episode(s) will include something about Dany being pregnant with Jon's baby, or with them holding their infant child. In GRRM fashion, John and/or Dany might die or be on their deathbed just before or after the child is born.
Note - I have completely different (though in some ways consistent) theories about how the books will play out.