What I mean, if Earth travels through space at 1,700 km/hour, when a spaceship or other object leaves its atmosphere, should it not stay behind? And if it is gravity that keeps it near Earth, should that speed not disintegrate them?
We've already covered that this is the speed of rotation at the equator, it goes much faster around the Sun. But anything launched from the Earth is starting on the earth and in the same state of motion as the Earth. One of the first things you learn in physics class is that a body in motion will stay in uniform motion unless acted on by an external force; in the absence of something like friction or wind resistance it will just keep going. Gravity is a force, of course, pulling inward toward the earth, and that's why the space station orbits the planet instead of flying away in a state of uniform motion. It causes every part of the space station to accelerate by the same amount and in the same direction, so there are no stresses that would cause it to disintegrate. (Footnote: there are small differences due to gravity being central and proportional to 1/r2, so the acceleration of gravity is not quite the same at every point on the space station. But that's in the microgravity regime, very small. You need some extraordinary conditions, like being close to a neutron star, for those so-called tidal forces to be a threat.)
I might add that rocketeers usually like to launch their rockets in the direction of Earth's rotation (east) because the rocket gets a boost to its speed. The rotational speed actually does matter. It matters for those involved with artillery, too; you can look up the Eötvös effect.
Short answer from someone who doesn't have a degree in physics and is bad at math.
While in space you are in a vacuum, since you are in a vacuum there is almost no resistance. When moving through the atmosphere there is air which provides resistance against the direction you are moving.
I say almost no resistance because at the height of the ISS there is still some resistance present therefore the ISS needs to boost itself once a while.
As for the leaving behind part. When in space you are almost always in orbit of something. This means that an object with an higher gravity is pulling you towards the object but since your speed is higher than the force pulling you you will stay in orbit around the object pulling.
please correct me if I'm wrong