How would your weight change with time if you were on s space ship traveling away from earth toward the moon

Well, before we discuss that, I think we have to carefully understand
and agree on something.  We have to be very clear about what we
mean by 'weight' ...  is it what you feel, or is it the product of

           (your mass) x (the acceleration of gravity where you are).

If you're on a space ship, then any time your engine is not burning,
you feel weightless.  It doesn't matter where you are, or what body
you may be near.  If you're not doing a burn, and the only force on
you is the force of gravity, then you don't feel any weight at all.

But of we say that your 'weight' is the product of

           (your mass) times (the acceleration of gravity where you are),

then it depends on where you are, and whether you're close to
the Earth or closer to the moon.  You may not feel it, but you're
going to have weight, and it's going to change during your trip
in space.

You know that the force of gravity depends on how far you are
from the body that's attracting you.

-- As you travel from the Earth to the moon, gravity will pull you
less and less toward Earth, and more and more toward the moon. 

-- Your weight will get less and less, until you reach the point
in space where the gravitational attractions are equal in both
directions.  That's about 24,000 miles before you reach the
moon ... about 90% of the way there.  At that point, your weight
is really zero, because the pull toward the Earth and the pull toward
the moon are equal.

-- From there, the rest of the way to the moon, your weight will
start to grow again.  It begins at zero at the 'magic point', and it
grows and grows until you reach the moon's surface.  When
you're there, your weight has grown to about 1/6 of what you
weigh on Earth, and it won't get any bigger.  If you weigh
120 pounds on Earth, then you weigh about 19.86 pounds on
the moon ... PLUS your space suit, boots, heater/air conditioner,
oxygen tank, radiation shielding, radio, and all the other stuff that
you need to survive on the moon for a few hours.