PLEASE READ THE WHOLE QUESTION AND THEN ANSWER
At what value of Force, could we feel or see the effect of gravitational pull or force between two objects.
E.g. 2 pencils kept at a very small distance (r) exert a force (F) on each other such that they are pulled by each other and stick to each other though the force is easy to overcome.
But actually, I want to see two bodies moving towards each other (ignoring Friction and Force of gravity due to earth on the objects).

You'd need to do something similar to the Cavendish experiment:

Cavendish experiment - Wikipedia, the free encyclopedia

I'm sure there are more recent methodologies that you could find.

Your feeling of a force or ability to sense a force does not depend on the nature of this force. It can be anything: gravity, electricity, inertia, etc. In the absence of friction, two pencils would move to each other all the time.