Newton worked out the whole thing with calculus, which was his private invention. Newton began with three laws of motion. Note that acceleration may be a change in speed, or in direction. A body's change in motion is proportional to the force acting on it, and will be in the direction of the force. The larger the mass of the object, the more force it takes to move it.
Every force has an equal and opposite force.
When you push the floor, the floor pushes back, and you jump off the floor. In addition, Newton described his law of gravity. The combined application of Kepplers laws and Newtons laws finally solved the problem of planetary motions. The predicted positions of the planets were in agreement with the observed positions - well almost. As the instruments of measurements became more refined, some very minor, but real discrepancies began to appear in the measured and actual positions of the planets - this was most evident with the position of Mercury.