The Doctor is reminded by Nardole that he made a vow to stay on earth and guard whatever the hell is in the vault which means no unnecessary travel. Given that the Doctor has a Tardis, this isn't really a big deal because he can arrive back moments after he left. I'm not sure that the Doctor had that in mind when he made that vow but with a new companion after so long, he simply cannot stay put and the chance to give Bill a peek at the universe is compelling to him.
Bill has gotten used to the idea that the Tardis is larger on the inside but that doesn't mean that she thinks that it makes sense or constitutes what she deems to be a proper space ship. For instance, why are the chairs so far away from the console? The tenth Doctor explained that the Tardis was actually meant to be controlled by six people, so perhaps the chairs are so far away so that one can get an actual break when a rest is needed. The Doctor also reveals that he stole the Tardis but if you recall in The Doctor's Wife, the Tardis claims that it was she who in fact stole him because she wanted to travel and see the universe. It's then that the Tardis reveals that she takes him where he needs to be which explains why wherever the Doctor lands, there's a problem which needs to be solved and why he sometimes misses the mark so badly when he tries to return to a specific point in time.
Bill like Rose chooses to go to the future first and akin to Rose is disturbed by the idea that humans have fled earth. Sure, Bill isn't seeing the end of the world; however, the very idea that humanity felt the need to leave earth is disturbing to her. Bill and the Doctor land on an new outpost built by humanity and find it devoid of life. The Doctor may not have been there before but he's not surprised by its existence given that he's run into human life preservers before in The Beast Below. Similarly to Ark in Space, humanity has made the trip in suspended animation, a factor the Doctor learns just before he can destroy the new habitat because of robots run amok.
The Doctor may change every time that he regenerates but some things always remain the same. The Doctor very strongly believes that he has a duty of care when it comes to his companions, even though he routinely places them in dangerous situations when it comes to his various adventures. It's certain that wherever the Tardis lands that there's going to a mystery which needs to be solved. Bill is initially amused by the robots who sport emojii faces. I think that there should be a shaming tax on the Japanese for burdening us with emojiis in the first damn place.