What happens when the drag force exceeds the weight of an object falling into earth? Let's say a meteor is coming towards earth. It's not accelerating, but it does have an initial velocity. This meteor is shaped so it has an insane amount of drag, enough to even exceed its weight (not mass) as it gets closer. What happens? Why? 
At first I thought it would just stop, but that doesn't really make sense. The forces cancel but that doesn't mean the body slows down. Would it keep going in another direction? 
 A: It would accelerate upward. This is exactly what happens when skydivers open their parachutes, for instance. 
The underlying confusion behind this question is likely due to mixing up velocity and acceleration. You can accelerate in one direction without also having a velocity in that direction: you can accelerate up while moving down. Hitting the brakes on a car is not the same as putting it in reverse.
A: The acceleration (or deceleration) of an object is $a$: $$a = f/m.$$
Acceleration (or deceleration) is the rate of change of velocity.  So, to find the rate of change of velocity of an object you divide all the forces acting on it, by the object's mass.
The forces acting on your hypothetical object are the object's weight and the atmospheric drag.  Someone being very picky might also say that there is a relatively very small atmospheric buoyancy at work too, but it can be ignored in most cases.
Let's say the object is falling straight down.  Atmospheric drag is velocity-dependent, with low drag at low speeds and much higher drag at higher speeds.  Drag will cause the object to decelerate, until drag equals the object's weight.  At that point the object is said to be moving at terminal velocity, and it just keeps falling at terminal velocity.  
The whole scenario gets more complicated when altitude-dependent atmospheric density is taken into account, but what's described above captures the essence of an answer to your question.
A: Drag is proportional to the object's velocity (or velocity squared, depending on regime), so as the object's speed decreases, the drag decreases as well. Therefore, the body won't start "falling upwards" because for that to happen, the velocity must reach zero, and at that point the drag also drops to zero.
In practice, the falling object slows down until the drag equals the weight, at which point it keeps moving at constant speed (so-called "terminal velocity"). In the hypothetical scenario where you have a constant force that's greater than the weight pulling on the object, then the object will indeed start falling upwards.
A: You've probably already seen what happens yourself with extremely small particles, such as those that comprise smoke.  Lots of meteors are this small.
At this scale, things still fall, of course, but it takes a long time.
If things get small enough, eventually Brownian motion takes over.
A: It will decelerate, yes. If the vector force exceed the vector force of the weight.
Note that both will increase as altitude decreases.  
If the drag exceeds the weight and speed becomes 0, then the velocity will become negative. (Think parachute in a thermal or strong updraught). It may be a pumice-like meteor or quite small.
If it is lighter than water then the weight will become negative when it hits water, so there is that to consider.
