We're not traveling across the Sun, we're revolving around the Sun, relative to the Sun that is.
And the Milky Way galaxy itself is rotating around the center of the Local Group. And on and on it goes!
I remember the Moon landing so well when Neil Armstrong and "Buzz" Aldrin landed the Moon capsule Eagle ( Houston, the Eagle has landed. By the way, Houston is the first word spoken on the Moon ). I was twelve years old and it was exciting times. I remember the adults saying things like, " We beat the Russkies, those commie pinko bastards! " and other such things. I was going to post a photo of Buzz Aldrin saluting the USA flag on the Moon but that would put me over the top on image postings for now so I'll do it later.