when you raise the minimum force it would be just off the ground, right?

Yes. You will get something like : The minimum force must be greater than ...$\displaystyle N$.
In fact, if you maintain the minimal force required to rise the body in the air applied on the body, it will rise up in the air faster and faster.
And about
So, since the force of the weight=mg and when you raise the total force it is in the upward direction, when you raise the minimum force it would be in the negative direction, right?

The answer is no.
Chose a referential system in which the positive direction is upward and the negative direction is downward. The weight (which is a force) has its direction downward while the minimum force required to raise the object must be upward, otherwise the sum of the forces will point downward (that is toward the center of the Earth) and the object would not move at all.
EDIT : If you don't mind, once you get the answer post it here so that we can check it out.