Newton formulated three basic laws of motion which laid the groundwork for classical mechanics in physics. We will go through each of them and how they relate to riding a roller coaster.
The first law states that, in the absence of any outside force, a body in motion tend to stay in motion and a body at rest tends to stay at rest. This is why the coaster needs to be mechanically pulled by a chain or track at the beginning since it is initially at rest when the riders get into the cars. Also, at the end of the ride, the coaster must have a braking mechanism to stop the cars for the riders to get out. It it true that friction would eventually stop the coaster, but since it is in motion an outside force (brakes) must be used to stop it effectively.
The second law states that F=ma, or force is equal to mass times acceleration. When the roller coaster is going down its first big downhill drop, the acceleration due to gravity is acting on it in a downward manner thus giving a downward force to the coaster. Also, if the coaster is full of people it will have a greater mass than an empty coaster. The greater mass of a full coaster will produce a larger force due to gravity's acceleration according to F=ma.
The third law states that if one body exerts a force (F) against another body, then the other body exerts a force (-F) on the first body of equal magnitude in the exact opposite direction. Since one would hope that there is no collision of any sort during a roller coaster ride, the only real example of this is that any force that the coaster exerts on the track, the track exerts an equal and opposite force on the coaster. So the track doesn't move against the coaster and the coaster doesn't move against the track (it moves along the track but not against it).