The solar system orbits the Milky Way galaxy at an average speed of about 828,000 kilometers per hour (515,000 miles per hour), taking approximately 230 million years to complete one full orbit around the galaxy's center. This journey is often referred to as a "galactic year."
If you could teleport instantly thousands of light years outside our galaxy and look down upon it with supertelescoping vision you might see something like this:
The Milky Way galaxy spins like water going down a drain with our local group in one arm halfway between the core and the outside edge at a distance of around 27,000 light years.
Our solar system orbits with the Sun dragging the other planets in tow, making the solar system look like it is flying sideways, always moving up since the solar system is tilted that way.