The number @Hawaii_Jake quoted is the speed of our sun relative to the center of the Milky Way Galaxy about 27,000 light years away.
You can easily calculate how fast our planet is moving around the sun. Say it is moving in a circle with a radius of 93 million miles and it takes year to make a full circle.
How fast is our galaxy moving through space is a little tougher. We can say how fast other galaxies are moving away from, or toward, us by looking at their red shift of light. Our nearest neighbor, Andromeda Galaxy, is actually moving toward us at 100km/sec! That is rare. Typically they are moving away from us at 1000 km/sec if near us, to 500,000 km/sec for distant galaxies.