This problem was first solved by Galileo, early in the 16th century AD.
IsItSo said:
If you put something in space and move it, it stays moving at the same speed. Earth itself is doing so at 500,000 MPH right now. If one of our satellites were to be placed in space at a exact location and so that it isn't moving AT ALL then it would keep its location in space while it watch Earth fly away from its-self fassst.
Indeed it would... If you really did place the satellite in space in such a way that "it isn't moving at all". But stop and think for a moment about what it means to place the satellite in space in such a way that "it isn't moving at all".
You are floating motionless in space watching the Earth zoom by at 500,000 mph. When you say something "isn't moving at all", you're saying that it is moving at zero mph relative to you. So yes, if we place the satellite somewhere in space where "it isn't moving at all" then of course it and the Earth will move apart at 500,000 mph - that's what happens when one thing is moving at 500,000 mph relative to you and the other thing is moving at zero mph relative to you.
But launching a satellite from a planet moving past you at 500,000 mph isn't placing it in space in such a way that it isn't moving at all relative to you. The satellite was moving at 500,000 mph relative to you while it was sitting on the launchpad... and as you have correctly said above, if something starts out moving "it stays moving at the same speed". There's no force acting on it to slow it down after it leaves the launchpad, so it keeps moving at 500,000 mph relative to you. And if the satellite and the Earth are both moving at 500,000 mph in the same direction relative to you, they're both moving at zero mph relative to one another, so they don't move apart.
Someone moving in the same direction at 200,000 miles relative to you would describe the situation differently: they would say that they were floating motionless in space while the Earth and the satellite (whether on the launch pad or above the surface of the earth) were zooming by at 300,000 mph and you were moving in the opposite direction at 200,000 mph. Both descriptions of the situation are equally valid.
Now try describing the situation from the point of view of someone who is moving at 500,000 mph relative to you. What do they say about the motion of the Earth and the satellite?