Does launching a space shuttle or rocket change the earth's orbit?
After all, to get momentum in space you need to throw something out.
Technically yes, but not in any way that matters.
Planets are enormous; Earth's mass is about 6,000,000,000,000,000,000,000 tons. All the spacecraft to ever leave Earth's gravitational sphere have probably totaled less than 1000 tons; we couldn't even measure the change in Earth's orbit produced by those launches.
In the case of the US space shuttle, and any low-Earth-orbit space mission, almost all the mass exhausted in launching the spacecraft returns to Earth in short order, so the orbital changes are canceled out in any case.
As long as "what happens in GEO stays in GEO" and there's no mass lost to deep space, the Shuttle and the Earth will mostly orbit around their common center of mass.
When the Shuttle is moving (relative to Earth) in the prograde direction (Earth's motion around the Sun), the Earth will be moving slightly slower than it normally would, but a half-orbit later when the Shuttle is moving retrograde, the Earth will be moving slightly faster than it normally would.
When the Shuttle returns, to 1st order (and maybe 2nd order as well) the Earth will be in the same place it would be if the Shuttle had never taken off.
It's quite a bit like the Earth and Moon orbiting around their common barycenter, and it being pretty much the EM barycenter that moves in a nice elliptical orbit around the Sun, except that as @RussellBorogove points out very effectively, it's a quite a bit smaller effect than that.