As far as I know we're headed toward another galaxy. Luckily we'll all by long gone by the time that collision happens.

“Where is the Galaxy taking us?”

Towards the andromeda galaxy which is over twice the size of the Milky Way. We are hurtling towards each other at about a quarter millions miles per hour.

For thousands of years after you die, that little fuzzy spot near Cassiopeia will slowly get larger and larger in the sky, and in about a four billion years, long after the Earth's oceans have dried up and the sun is a giant, reddish monster hovering in the sky, and our magnetic field will have long since died out, our atmosphere will have been mostly stripped away and the weather will feel like being on the highest mountains in an oven, the night sky will be covered with a dazzling display of the Andromeda galaxy overhead, spiral arms visible with the naked eye stretching from horizon to horizon.

We will merge, in a series of passes through each other, with almost no stars actually colliding most likely, although a good number will be ejected into the emptiness of intergalactic space, and will finally settle into a new shape, and may trigger a new phase of star formation as new clouds of gas and dust collide and collapse in the new super-galaxy.

Fun fact, we do not just orbit the galaxy in a circle, we also have a motion perpendicular to that circle. We oscillate up and down through the plane of the Milky Way. The Milky Way is super thin, like super ultra thin. If the Milky Way were a pancake, it would only be the thickness of a sheet of paper, a sad pancake indeed. However in terms of human scales it is still huge, so we have a large way to travel. Our galactic orbit is tilted as compared to the galactic plane, so throughout the cosmic year we move up and down as compared to the center. A motion of 100-200 light year, so pretty big. That orbit also has procession, so we move through different parts.

The galaxy itself is also moving, although at that scale it's easier to think of the galaxy to be stationary and other galaxies moving towards or away from us. In general we are all moving towards a galaxy cluster known as "The Great Attractor" as it is the most massive (except for your mom).

It's also often forgotten that our sun isn't the only star moving in the galaxy. All of the stars orbit the galaxy in a lot of different orbits. And some don't orbit at all, instead moving with escape velocity (or faster) to get flung outside of our galaxy. Some have their own orbit in companion dwarf galaxies that in turn orbit our own galaxy. It's easy to think of a galaxy as a fixed thing, with all the stars in the same place moving together like on a disk. But this isn't the case at all, stars aren't bound together and can follow their own path. Over time their relative positions change and the constellations we know won't exist anymore.

The structures we see in galaxies like spiral arms for example are only structures in the same way a wave in the ocean is a structure. It is clearly a thing that exists, with properties we can at least somewhat constrain (like size for example). But the water inside that wave is just water like everywhere else. At one point it's part of the wave and then at some point it no longer is. It's the same for stars, sometimes part of a structure, other times not (although it gets complicated quickly if you dig into the details)

The galaxy is taking us to see its friend

Now do the local group and super clusters! The universe is wild!