What is the effect of air resistance on the acceleration of falling objects?

All objects want to stay where they are. This is a simple rule of physics. It is called Inertia. When an object passes through the molecules that are in the air, the air particles, which want to stay where they are try to force the object that is moving, or indeed as you mentioned; falling to stay in its position. Because air can not do this, you get the effect which is called terminal velocity - the speed at which an item can travel at it's maximum through our atmosphere.

The effect of air means that falling objects can only hold up the constant acceleration due to gravity for a certain period of time before the object stops accelerating.

Another interesting effect that objects falling at high speed heat. This is due to the object that is moving smashing into the air molecules. This transfers energy and causes heat.