Arya's immediate future.
Arya is currently at a loss, directionless with a hole inside her in place of a home. She longs for her wolf pack and loves her wolf dreams, signalling her loneliness as she completely lacks for relationships, friends or family. Sansa will go from the Vale to KL, it is Sansa's arc to play the game of thrones and KL is where it is played. Arya will join with her there, though Arya will not reveal her identity to Sansa she will become her cat's paw, that's what the FM training is for, and why GRRM said Arya won't necessarily be assassinating a heap of people back in westeros. Collect information and bring it back to base. She will be Sansa's eyes and ears in KL, doing as she's told, the leg work, providing her information and carrying out her plots as Sansa plays the game of thrones and works her way up.
Arya will at first be happy. Sansa will provide her direction and a place, and they'll be bringing down the enemies of House Stark, vengeance. Likely part of what Arya will do for Sansa is kill and assume the identity of Nymeria Sand so that she may infiltrate Aegon's small council.
Whatever the reason Arya will head to the Riverlands and to Stoneheart. Perhaps a mission from Sansa to clear it out so that she can manouver to have Edmure reinstated. Again Arya won't reveal her identity. She will observe Stoneheart and come to understand that it doesn't have the capacity for love, and without the capacity for love it's not really her mother. Stoneheart's incapacity for love will also mean she is merciless. Eventually Arya will reject Stoneheart's blanket justice and vengeance and help destroy her. The tipping point could be a judgement against one or both of Elmar Frey and Jeyne Westerling. Gendry will be part of the faction that helps destroy Stoneheart and impress with his bravery which will be witness by Arya, Brienne and Jaime.
Through the Stoneheart arc Arya will learn many things. The difference between vengeance and justice will become clearer, and importance of the differentiation. How important it is that the crown does justice, and the horrors that can occur when the common people lose faith in the crown to provide justice and turn to other means. Where a path of anger and hate leads. From where mercy originates.
Sansa will continue to manoeuvre in the background during the dance of dragons, which Dany will win. With Arya's help Sansa will play a part in orchestrating a popular rebellion by the people of KL against Dany. Possibly Arya assassinates the High Septon, as he will be in confrontation with Dany and Dany will be assumed the culprit. And/or she may take a role as a street preacher and preach against Dany, her foreign forces, gods and dragons. This will tear Arya in two, though she will do Sansa's bidding she will like Dany a lot and Sansa's campaign against her will not sit well.
In the aftermath of Dany being forced to flee KL the Faith will hold power in the city, and they will call a Great Council to decide the next king or queen. It will come down to Cersei against Tyrion and Sansa, and Arya will help Sansa win.
Eventually Arya is going to have to confront a hard truth, Sansa isn't in this for justice or any altruistic reasons, Sansa is playing the game of thrones for the sake of Sansa's own power. Arya will not have revealed herself to Sansa, and having won the throne Sansa will probably think it'd be in her best interest to kill off her cat's paw. It is LF's lesson, dead men tell no secrets. So probably Sansa tries to have cat's paw Arya killed but Arya survives. Anyway, however it comes about Arya will eventually abandon queen Sansa and leave for the north where KITN Jon now rules.