Why does two objects with different weights fall at the same time, taking air resistance to be negligible? For instance, since weight is given by mass x gravitational field strength and assuming both objects have the same gravitational field strength and the only changing variable is its mass, why do they fall at the same rate?
 A: Here is the easy way to think about this.
We remove all the air, to make things simple. Hold your breath while we do this experiment:
Take stone in hand and drop it. Notice its time of fall.
Now hit the stone with a hammer to break it into two equal pieces. Carefully fit the two pieces together, and then drop it, noticing its time of fall. It falls to earth in the same time it did when it was only one piece. Why?
Because when it was one piece, gravity was acting equally on every little fragment of the stone- equally on the two halves before you broke them apart- and gravity doesn't care about the chemical bonds acting between the silicon oxide units that make up the rock. All the atoms in the rock experience the same gravitational field, and so all portions of the rock fall to earth in the same time.
A: The heavier object takes more force to accelerate but gravity exerts more force on it since there is more mass to act on. The lighter object takes less force to accelerate  but gravity exerts less force on it since there is less mass. The result is that it balances out so they have the same acceleration. That is to say, the force of gravity acts on a per unit of mass basis, not on the basis of the mass of the entire singular object, whether it be two different heavy and light objects, or a single heavy object or the same object split into two pieces.
You already know that it takes more force to give a heavier mass the same acceleration, and you can see from the gravitational force equation that the force exerted is larger when either the planet's mass or the object's mass is larger:
$$F=G\frac{m_1 m_2}{r^2} = \left\{ G\frac{m_1}{r^2}\right\} m_2=m_2a$$
And if we plug in the gravitational constant, Earth's mass, and Earth's radius, we get
$$ a=\left\{ G\frac{m_1}{r^2}\right\} = 9.81m/s^2$$
So the object and the planet exert the same force on each other and both accelerate towards each other. But since the mass of the planet is usually so much greater than that of the object, that same force accelerates the object much more than it does the planet so the planet appears to remain stationary with the object moving toward the planet rather than them moving toward each other.

"assuming both objects have the same gravitational field strength"

This can't be true if they have different masses.
