Question

What direction is our galaxy traveling?

Answer 1

There are lots of spinning objects to consider and the angles between their axes vary over time, but we can answer the question for where the Sun currently is in it's orbit around the Milky Way.

The sun moves around the Milky Way in a clockwise direction (viewed from 'above' galactic North) and the ecliptic (plane of the Earth's orbit about the Sun) is angled at about 60 degrees to the galactic plane.

Answer 2

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It appears to be moving towards our Sun at a speed of 301 km/s, which means —when we factor in the motion of the Sun through the Milky Way — that the local group's two most massive galaxies, Andromeda and the Milky Way, are headed towards each other at a speed of around 109 km/s