I'm trying to understand gravity in General Relativity and I'm having some questions. I can understand that an object in orbit around another more massive object is free falling and simply following a geodesic. What I can't understand is, if those objects are standing still relative to each other, why would they ever "start moving" along a geodesic until they collide? My feeling is that the problem is somewhat related to my definition of "standing still", which in this case it would be that the objects pop up spontaneously out of nowhere without any force acting on them and they happen to be close.
my definition of "standing still"
Exactly - you are not standing still in four space. Just look at the hands of a clock!
I always liked this one:
the reason you're sitting in your chair is that the shortest distance between today and tomorrow is through the center of the Earth.
You must remember that objects are never "standing still" in general relativity, as the four-velocity in spacetime is always nonzero for any object. The value of this four vector is what defines the initial condition for geodesics, not the three-velocity in any one reference frame.
"Spacetime tells matter how to move; matter tells spacetime how to curve." - John Wheeler
Anywhere there is mass/energy, space time is distorted. However, a small mass distorts it less than a big mass. Geodesics explain how small masses move in a background geometry. However, they do not account for how masses affect the geometry itself.
In your scenario, you do not want to neglect the mass of either object. Therefore, in order to understand "why" the masses attract, you would need some understanding of the full Einstein field equations. This is more complicated than small masses moving along geodesics.
In other words, you need to not only understand that "spacetime tells matter how to move." You also need to understand how "matter tells spacetime how to curve."