In Game of Thrones, it is shown that the Three eyed Raven (aka Bran) is capable of seeing the past through the eyes of any person in the human history. So why couldn't Arya simply ask Bran what's "west of Westeros" instead of sailing into the unknown? Surely Bran could relegate the story of every single human settlement on the surface of the earth?
1. We've no evidence that Bran knows anything about anywhere other than Westeros
This is obviously not definitive, we know essentially nothing about the limits of Bran's abilities. But it's possible that this is a question he just doesn't know the answer to, or may decline to answer for reasons we couldn't even begin to guess at.
2. Arya isn't looking for an answer, she's looking for something to do
This is the big one. Arya's story is mostly about revenge and independence. Her enemies are dead, and no one can make her do anything she doesn't want to do.
What other options might seem compelling to her? She's not going to be to be the Lady of Winterfell, even if she wanted to be. She seems uninterested in playing the role of Westerosi noble at all. The only trade she knows is fighting and killing, which she can do at her whim at any time. She seems to have had enough of being a contract killer back in Braavos.
But whether or not she's looking for adventure or something else, it seems unlikely that she's trying to satisfy some nagging curiosity about the literal contents of regions West of Westeros. She's more of a roving loner, like Nymeria, so she's off to rove.
I agree with Upper_Case's answer on why Arya doesn't care about it, but think this is also simply not possible for Bran. While we do not have an exact definition of Bran's powers it seems many people make two assumptions that are not correct, one being he can see everything everywhere at will and the other being he can see in the future. The latter is not terribly important here, so I'll focus on the former.
As a reference, let's start from the books, there is no three eyed raven but the similar concept of a greenseer. These are powerful 'mystics' that possess both the greensight (vague 'prophetic' dreams), are powerful wargs (animal posession), and supposedly could see current events through weirwood trees. For a bit more explanation see here and here.
In the show they turned the dial up to eleven. There supposedly is only one three-eyed raven at the same time, the powers seem to be transferred somehow from 'master' to 'apprentice' and three-eyed ravens seem to have the possibility of extreme longevity (though that may only be true in certain circumstances). As to the powers things are a bit fuzzy, but this is what we've been shown:
- He cannot simply see 'everything'. All events he sees (past or current) are linked to people, not places or other creatures. Most notably he still uses ravens frequently to do basic scouting.
- While he apparently has access to apparently all memories of all people, he needs some direction and guidance, he does not know everything. He needed Sam's input to start on the path of Jon's lineage but given that could quickly find what he needed.
So while Bran is very powerful and seems even more powerful than in the books, it is also fairly limited. He cannot just put his mind to the west but would need to connect to some (supposedly known) living person or memory of somebody who went there, but apparently they do not know of such a person. He cannot also just see it but would need to use e.g. ravens or crows to scout, which would take a lot of time, energy and the birds may not even survive.
I also want to point out the 'future' seeing again, I strongly believe he can do no more than make very good guesses based on his rich knowledge of the past and present, possibly complemented with greensight prophecies that he has no control over. The proof for that is that he did not expect Arya to come north but keep heading south for Cersei. This also avoids a paradox in this case where he could go into Arya's future to check west of Westeros out but that would invalidate her quest (aside from her adventurous spirit).
Bran's visions (the ones we know of) have always been in the vicinity of weirwood trees. There is a strong connection between weirwood trees and the old gods (which is where the greensight comes from).
Most of the weirwood trees in the south of Westeros have been chopped. This is mention in the first (or second) season of the show. We also don't see Bran having any visions in the South of Westeros, except for two occasions: Jon's birth and Rhaegar and Lyanna's wedding. Both happened right next to weirwood trees.
So the logical answer is that Bran doesn't know what's west of Westeros because he there are no weirwood trees there.
As a minor caveat, you could argue that if anyone discussed what's west of Westeros in front of a weirwood tree, Bran could find that out. But he has been shown to not know everything yet. He simply has access to everything, but he needs to know where to look.
This is why Bran doesn't know to look for confirmation of Rhaegar and Lyanna's wedding until Sam points it out to him, at which point Bran is able to focus on the right time/location to recall the information.
We've no evidence that Bran knows anything about anywhere other than Westeros
Well, from the first book we know
"Finally he looked north. He saw the Wall shining like a blue crystal, and his bastard brother Jon sleeping alone in a cold bed, his skin growing pale and hard as the memory of all warmth fled from him. And he looked past the Wall, past endless forests cloaked in snow, past the frozen shore and the great blue-white rivers of ice and the dead plains where nothing grew or lived. North and north and north he looked, to the curtain of light at the end of the world, and then beyond that curtain. He looked deep into the heart of winter, and then he cried out, afraid, and the heat of his tears burned on his cheeks."
And he saw it even before waking up for the first time after his fall. So, he knows about other places than Westeros
I'm agree that Arya doesn't want to be told about it, she wants to see it, she is looking for adventures and a good fight. She is just that kind of person. Even when they was heading to the King's Landing with her father, she managed to find them every day, she enjoy it