Which is heavier, the earth or the moon

When you say "heavy", you're talking about the gravitational force
between that object and another object, so it depends on what the
"other object" is.

If the "other object" is, let's say, the sun, then the gravitational attraction
between the Earth and sun is about 80 times as much as the gravitational
attraction between the Moon and sun, because the Earth has about 80 times
the mass of the Moon.

But if, somehow, the weight you have in mind is the gravitational attraction
between the Earth and the Moon, then those forces are equal.  The force
of gravity between two objects depends on the product of both masses,
and it's equal in both directions.

If that isn't clear to you, let me give you this additional fact that's guaranteed
to knock you even further off-balance:

Your weight on the Earth is determined by the product of

                           (your mass) times (the Earth's mass).

The Earth's weight on you is determined by the product of

                           (your mass) times (the Earth's mass).

Your weight on Earth is the same as the Earth's weight on you.

Would you like to prove it ?

-- Turn the bathroom scale upside-down, so that the step-pad
    is on the floor.

-- Then step on it, so that you're standing on the bottom, which
    is facing up.

-- If you placed a little mirror on the floor, so that you can read
   the numbers, which are facing down toward the floor, you'll
   read your own weight, even though with the scale upside-down,
   you're weighing the Earth on you.