Someone!

how does an increase in distance between objects affect the gravitational force?

Well I don't know.  Let's think about some of the things we know about the solar system, and see if we can reason it out.

-- Jupiter is the biggest planet in the solar system.  It's about 11 times the size of the Earth, but we don't notice any forces pulling us toward Jupiter.  If we jump out of an airplane, we fall to the Earth, not to Jupiter.  Jupiter is always between 391 million and 577 million miles from the Earth.

-- The sun is about 8,000 times the size of the Earth, but we don't notice any forces pulling us toward the sun.  If we jump out of an airplane, we fall to the Earth, not to the sun.  The sun is always about 93 million miles from the Earth.

It sorta looks like even ginormous huge things don't have much gravitational force on us if they're much farther away than the Earth is.

That impression is correct.  The force of gravity decreases with the SQUARE of the distance between objects.  For example, if you move the objects twice as far apart, the force of gravity between them drops to 1/4 of what it started out.  And if you move the objects TEN times as far apart, the force of gravity between them drops to only 1% of what it started out.

Closer together ==> more force of gravity

Farther apart ==> less force of gravity

Extreme example:  Take the Andromeda GALAXY.  It has BILLIONS of stars in it, and as much mass as BILLIONS of BILLIONS of Earths.  But it's so far away that we don't notice any gravity force at all pulling us toward the  Andromeda galaxy.

wvbbvvv