A question about space and space travel...?

How does the shuttle change direction and speed in space when there is no air for the gasses to push off of?

"for every action there is an equal reaction" by burning fuel of kind, they create a backward push, irrespective of the surroundings, the shuttle will go forward

It is not correct to suppose that rocket exhaust gases need to push against anything. The fact that they are exhausted is all that is necessary, according to Newton's third law, to cause a change in velocity.

A shuttle will use fuel to get out of the earth's gravity field and then, if everything goes to plan, it will drift in a certain direction until it gets pulled into the gravity field of another planet and then it will 'fling' into another direction using the gravity like a slingshot. This is planned before hand so the shuttle will usually not have to use fuel once out of earth's gravity field because it will drift in the right direction because of where the shuttle is respective to the planets it needs to use. If the shuttle goes off-course or is going too fast or slow it can use fuel to 'blast' itself into the right direction.

All of these are correct.

Newton's laws of mechanics do not require anything "to push against". That is a common misconception. One way to see how this works is to view the mechanism as one that obeys the principle that the center of mass does / must remain "in the same place" before and after some mass is moved from place to place.So if we start with a rocket ship in outer space, in a vacuum, and start throwing out particles in one direction (that is, fire up the booster rockets) the vehicle has to move in the opposite direction so that the center of mass remains fixed in the same place that it started. The more particles we throw out, and the faster we throw them, the further and quicker the vehicle must move in the opposite direction to keep the center of mass unchanged.This is a way of thinking about the propulsion process, and is an oversimplification, but it is basically true. It is one of the conservation laws, the conservation of mass. There are other conservation laws as well: energy, momentum, angular momentum that must also be obeyed too, but the of the center of mass is sufficient for your inquiry.