Short answer: No, the sun is not stationary.
Along with the Solar System that includes planets like Earth and Jupiter, as well as asteroids, comets etc. associated with the sun, it moves around the center of the Milky Way Galaxy.
The apparent motion of the sun that one sees daily (if it is not cloudy) is due to the rotation of the Earth. It is because of the apparent motion of not just the sun, but also of stars at night (I know, if its is not cloudy), that until several centuries back it was thought that the Earth is the center around which everything else moves.
That theory could not explain certain phenomena (e.g. phases of Venus). The center then shifted to Sun with all Solar System planets (and asteroids, and comets) going around it.
Further observations of the night sky revealed that all stars are somewhat like the sun (though there is a great variety in them). There are billions of stars in our Galaxy. Our sun is at the edge of the Milky Way Galaxy, and the entire Galaxy rotates roughly around the center of the Galaxy.
To take a not completely accurate analogy, think of the Galaxy as a merry-go-round. If you sit on one of the seats, and hold a smaller model of the merry-go-round in your hands, that model would be like a Solar System with its planets going around the center, and the small merry-go-round going around the center of the Galaxy. [Can you identify the inaccuracies in this model?]
The Milky Way Galaxy too is just a galaxy, and there are billions of those too. The galaxies themselves are moving away from each other at great speeds.
Motion, and stationarity, are concepts that one can define in context of a specific frame of reference. Despite the different motions that the Earth is involved in (around itself, around the Sun, around the center of our Galaxy, and away from other galaxies as part of the Milky Way Galaxy), for most practical purposes (e.g. to go from your home to your school or workplace on a given day) you do not have to consider the motions of other planets, stars and galaxies. But when we launch spaceships, the movement of planets is important because we can make use of their Gravitational force (the only force that works at long distances) to speed up our spacecrafts and direct them where we want to send them.
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