I should make clear GR isn't my thing. I've played with SR a bit and claim some 'feel' for spacetime - my attempts to introduce acceleration into SR haven't gone well. The accuracy of what comes next is best judged by the reaction of others. The first experiment should be to go to the top of a high building while holding a cricket ball. Jump off and gently push the ball away. Observe the path of the ball ... (ignoring air resistance) it travels in a straight line. The mystery isn't so much what the ball is doing as why the street below is rushing up to meet it. The street is rushing up to meet the ball because the street is prevented from falling towards the centre of the Earthy by all the stuff underneath it. If I were rash I'd say the parabola associated with projectile motion is the result of viewing an inertial object (the ball) from an accelerating frame. You could ask me why I think the presence of mass affects spacetime in any way and I would have to admit to having absolutely no clue whatsoever.