The speed of light is supposed to be the same for all observers in an inertial frame, no matter the observer's speed. As a result, time slows down for observers moving quickly, and this explains why light always seems to move away at the same speed. However, what if light moves toward a person in an inertial frame traveling at, say, half the speed of light? Without time dilation, the speed of light would appear to be be moving at a speed greater than the speed of light, in the observer's frame. The only way to fix this problem is for time dilation to occur; HOWEVER, to fix this problem, time should speed up for the observer, not slow down. And yet, for observers moving quickly, time slows down (which solves the problem of light moving away from an observer). Where is my misunderstanding in all of this/ what is going on here?
What you are thinking about is that time dilation is relative.
Let's say that in your case there is an observer moving at the c/2. OK, but compared to what? There has to be a observer at rest. So the observer moving at c/2 is moving relative to the observer at rest.
Now the observer at rest has a clock, and this observer at rest will see its own clock tick normally. But if he compares his clock to the clock of the observer that is moving at c/2, he will see that the clock of the observer at rest is moving faster then the clock of the observer moving at c/2.
The observer moving at c/2 sees its own clock tick normally. But when he compares it to the other's clock, he will see that the clock of the observer who is moving at c/2 is ticking slower then the clock if the observer at rest.
This is due to time dilation.
Now as per SR, all observers will see light moving at speed c (when measured locally, in vacuum), regardless of the speed of the observers or their direction.
In your case, it does not matter whether the light is coming toward the observer or moving away from the observer, and it does not matter whether the observer is moving toward the light or away from the light. The observer moving at c/2 will always see light moving at speed c.