Well, the sun is moving already. I assume you know that, but to summarize, our sun orbits the center of the Milky Way Galaxy, that’s called a cosmic year, and it takes about 225 to 250 million Earth years. So if a billion is a thousand million, every billion years the sun goes around the center of the Milky Way Galaxy 4 times.
The Milky Way Galaxy itself is moving.
Movement is a weird thing, because movement is always in relationship to something else.
The Milky Way and Andromeda galaxies are going to collide at one point. It’s unlikely that will mean any actual stars collide, because they are so far apart in any one galaxy.
With all this movement going on, the Earth just happily keeps going around the sun. If we propose the earth is 4.5 billion years old and the sun goes 4 times a billion years around the center of the Milky Way, that means that during the history of the Earth, our sun has orbited the Milky Way galaxy about 18 times.
Give or take.