How does a space shuttle accelerate in space?

Rockets don't "propel" against anything.  They don't need an atmosphere to function.
Newtons Third Law - "for every action there is an equal and opposite reaction" - is what makes rockets work.
The hot exhaust exits the rocket at high speed moving backwards, which causes the rocket to more forward.

Atmosphere is not required to accelerate.  The hot gases of a rocket exhaust provoke a opposite reaction (Ala Newton's law)
in the body of the rocket.  It gets thrust in the opposite direction.  In fact, you'll get more acceleration outside the air as you don't have to contend with the drag induced by the rocket moving through the air.

It's not that that puzzles me, Erik - it's how they manage to land back on earth with NO POWER EXCEPT BLOODY GRAVITY !!  This feat alone earns the Americans my deepest respect and ready acceptance as No 1 in the world!!

Einstein's E=Mc^2

In space, any mass can be moved by releasing directional energy. And energy can take many forms, the fuel of a rocket, gravity, the impact of two objects, solar rays, etc.

Perhaps the simplest example is to picture two astronauts floating in space. If they want to accelerate away from each other, all they have to do is push each other with their hands, and they will go flying off in opposite directions.

A rocket or shuttle uses fuel to create the energy required. In earth's atmosphere more fuel is required for an equal amount of velocity to overcome the friction caused by air.

The shuttle can also take advantage of the Earth's gravity to increase its velocity once in orbit, because gravity is also a  form of energy.

Reverse the question and question your initial assumption. The shuttle DOES accelerate and propel itself in space, therefore, perhaps it doesn't NEED to push against an atmosphere to do so. Moreover, satellites ever since the first rockets to outer sace have been able to propel themselves in the vacuum of space.

Rockets do not work by pushing against the air. This has been known since before even the beginning of the space age. The shape of the engine nozzle forces the expanding exhaust gases out in one direction, causing the rocket to move in the other in accordance with Newton's thrid law of motion. They actually work better in a vacuum due to the lack of atmospheric drag.