All motion is relative means just that. Things move relative to each other. You ask if the Earth might be stationary. Stationary relative to what?
You could, for purposes of whatever problem you are trying to solve, define the Earth as your frame of reference. We often do this when trying to solve a local problem, such as understanding the path a ball takes when we throw it.
If you define the Earth as your frame of reference for calculating how to get to the moon, it turns out the math is really complicated. As others have noted, at that scale the earth is an accelerating frame of reference. It is much easier to use a frame of reference that isn’t accelerating. So for going to the moon you probably want to define the Sun as stationary and use that frame. But if you are trying to go to another star, then you want to define our galaxy as our frame of reference.
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