Warning: Potential spoilers ahead for Game of Thrones! Read at your own risk...
The two were kind of at odds when they first met, but things have evened out in the episodes sense. Jon has shown Dany just how real the white walker threat is, she's allowed him to mine her mountains for dragonglass, and the duo have found a mutual respect for each other.
Unfortunately, Jon won't be staying in Dragonstone for much longer. He knows that there are multiple wars to be won, and he doesn't have any dragons at his side. In a perfect world, he and Dany would take all the armies they have straight to the wall, but Dany isn't going to let that throne out of her sight.
Jon is a mover and shaker, and he will need to get back to moving if he wants a chance at building his army and defeating the white walkers.
The question remains, where does he go next?
The most obvious theory here is that Jon Snow takes his mountain of dragonglass and heads back home.
Things were rocky when Jon left the North. Many of his people questioned his meeting with Dany, and some felt that he was turning his back on the folks who had just crowned him king. He left them all behind, promising to meet with Dany and return.
In order to keep his people happy, and to regroup, Jon could head back to Winterfell. This would be the best move if he's aiming to keep the North united.
There's also the issue of the dragonglass. Yes, ultimately that needs to find its way to the wall, but he could take the time to create an arsenal of weapons while hanging out at home.
It may not make all of his people happy, but Jon could skip Winterfell altogether and head straight up to the wall.
See, Jon has two things that are vital to stopping the Night King at the wall. He has a ton of dragonglass, and he has information.
That dragonglass needs to get in the hands of the Night's Watch and the wildling army as soon as possible, and Jon knows that. He also knows that telling them of Dany and her dragons will go a long way to helping everyone believe they can defeat the white walkers.
After leaving Dragonstone, Jon could head to either Castle Black, where he called home for many years, or over to Eastwatch-by-the-Sea, the castle where Tormund and his army reside.
The next episode of Game of Thrones is titled "Eastwatch," so thinking that Jon will head that way is a pretty safe bet.
This might be the least likely of all three options, but don't put it past Jon to do the unthinkable.
There's a chance that Jon could head south, right toward King's Landing.
If Jaime is alive, and Dany takes him prisoner, she could use that against Cersei. Perhaps Daenerys could suggest taking Jaime down to broker a deal, but Tyrion would talk her out of going herself. In that case, Jon would be suggested as a possible stand-in, proving his loyalty.
Jon could also head to King's Landing on his own accord. He needs all the help he can get in order to stop Winter from killing all of Westeros. There's a chance he could try and use Tyrion and Jaime to get a peaceful audience with Cersei and attempt to plead his case.
No matter which direction he travels, Jon Snow is likely heading far ways from Dragonstone in the near future.