I'm a teacher in Belgium and get a lot of questions about the universe. I did my research but never found out one thing which is also my question. Is there proof that the whole Galaxy (the earth, sun, moon everything) also travels with a certain amount of speed?
There is indeed evidence that the whole galaxy, and indeed the whole of the Local Group of galaxies (the small cluster of galaxies that the Milky Way is part of) is moving relative to the rest of the universe. What we use is the Cosmic Microwave Background. This is radiation left over from the very early universe and is very smooth (the variations are less than 1 part in 100,000). Since it comes to us from everywhere on the sky equally it makes a good reference to measure our speed against, and since it is very smooth you can see even quite small velocities. If you look at this image the big gradient that you can see is due to our motion through the Universe, the radiation seems slightly hotter in the direction we are moving towards and slightly cooler in the opposite direction due to the Doppler effect. Now of course Earth is orbiting the Sun, and the whole Solar system orbits the centre of the galaxy, but we know what these motions are and they don't account for all of the motion in that image, so the Milky Way must also be moving.