I've taught high school physics for almost thirty years. "g" is one of a
host of topics/ideas that kids have problems with. Otoh, kids have heard
about "gravitational fields" and have a fuzzy idea about what they are. It
doesn't bother them if you use the term. I introduce g as the strength of a
gravitational field due to the Earth, remind them that it varies with
distance, etc, and has a value of 9.81 N/kg at the Earth's surface. All of
this happens in the Newton's Laws topics where we have to use weight as the
force the Earth exerts on an object, which, in turn, occurs when we're doing
free-body diagrams. They are beginning, at this point, to get the idea that
mass and weight are not the same, and that weight depends on the strength of
the gravitational field, but mass does not.
Kinematics comes earlier in the course, but I do NOT do free-fall at that
time, so g doesn't come up prior to forces. Later in the year, I do
free-fall as a preliminary to projectile motion, and at THAT time we point
out that n/kg is equivilent to m/s^2, and can ALSO be used as an
acceleration. I think this sequence has a couple of advantages: The first
is that students see that g has two interpretations. The second is that it
sets up the concept of OTHER field strength terms being force divided by the
characteristic being influenced by the force (mass for gravity, charge for
electricity, charge*velocity for magnetism). The third is that free-fall
coming later gives them a refresher on kinematics (just as energy gives them
a refresher, and an alternative approach, related to free-fall). I find
that doing things that way unlinks g from purely acceleration
conceptualization, without reducing the appreciation for g AS an
acceleration. Seems to work pretty well.