When the Apollo astronauts were half-way to the moon (~130,000 miles
out), what force of gravity, in kg-m per second squared in what
direction, did they experience?

First off I found this at BadAstronomy.com:

"Bad: Another kid asks, "Can you hit a baseball to the Moon in space?"
and she says, "Yes; you just need to knock it halfway there, about
100,000 miles, and the Moon's gravity will take it from there."

Good: That's not correct; the Moon's gravity doesn't take over halfway
there. The Moon is a lot less massive than the Earth, and so its
gravity is correspondingly less. In other words, the weaker gravity of
the Moon means you have to be much closer to the Moon than the Earth
for the gravity to balance. The Moon has about 1/80 the mass of the
Earth. If you go through the math, you'll find that halfway between
the Earth and Moon (which she correctly gives as about 100,000 miles
away) the Earth's gravity is still 80 times that of the Moon's.

If you calculate where the gravity just balances, you'll find that it
goes as the square root of the mass ratio. In other words, you have to
be the square root of 80 times closer to the Moon than the Earth,
which is equal roughly to 9. That means you need to be about 216,000
miles from the Earth, or about 24,000 miles from the center of the
Moon. That's a long way to hit a ball!
"
http://www.badastronomy.com/bad/movies/spacecowboys2.html

So we know that relatively speaking, the gravitational force will be
80 times stronger from earth's direction that the moon's.

The problem appears here:
http://www.sfu.ca/phys/100/lectures/gravity/GravityExamples.html

They say that the force acting on a 100kg man is 1.1 N from earth, and
0.013 N from the moon, which means a total of 1.087 N from earth's
direction.

But you are after Kg m/sec2 - I'll leave that for someone else to work
out. The type of math you need to use is on this page:
http://users.erols.com/richdoran/gravity

The word "experience" is the key.

Within their moving frame of reference, with a constant velocity, the
astronauts "experienced" zero gravity. This is a practical example of
Einstein's elevator thought experiment.

http://spacelink.nasa.gov/Instructional.Materials/NASA.Educational.Products/Microgravity-Fall.Into.Mathematics/Fall.Into.Mathematics.pdf

hlabadie-ga

"They say that the force acting on a 100kg man is 1.1 N from earth, and
0.013 N from the moon, which means a total of 1.087 N from earth's
direction.

But you are after Kg m/sec2 - I'll leave that for someone else to work
out."

Just a really minor comment:
By the definition of Newton unit - "Kg*m/sec^2" IS Newton, so there
are no further calculations here.