What you are describing are two complimentary descriptions of the same process. You may either find it mathematically profitable to focus on the geometric properties of spacetime and the shapes of the 'straightest lines', or you can use the Newtonian description of forces. A falling body moves along a curved path because a gravitational force is present which changes its speed and direction at each instant, producing an acceleration. In relativity, a massive body warps spacetime in its vicinity in a precise way that depends on its mass, so that straight world lines in 4-dimensions become curved trajectories in 3-dimensional space which we see as ellipses or other 'conic sections'.