If the moon is slowly moving away from earth, what will happen to the gravitational force between the earth and moon? why?

There is a physical priniciple called the convservation of momentum, which says that the Earth will have to slow down as the moon gets farther away. This means that the length of the day increases slightly - more hours in a day is probably all you wanted! However, the increased day length is a tiny tiny fraction of a second in our lifetimes.

The moon wll never go away though. After billions of years (The precise number is hard to estimate), the moon will get to its final end state where it orbits more slowly and the earth apins exactly as fast as the moon orbits. One side of the earth will see the moon fully up in the sky while the other side of the earth will never see the moon. The state is called tidally locked. By this time, the sun may have already become a red giant and distributed or destroyed the earth and moon entirely, though.