I know my logic must be wrong but I can't figure out why. I know that charges must accelerate towards lower potential energies simply because that's a general rule of nature. However, when you release a positive and negative charge close to each other, they will move towards each other, and the distance between will decrease causing the potential energy to increase (1/r^2 relationship). So aren't the two charges gaining potential energy? Does it have to do with the sign of the charges being different? I feel like that explanation doesn't fit entirely because it seems like the magnitude of the potential energy should not be increasing.
The potential energy between two oppositely charged particles goes like $-1/r$. First of all, the potential energy doesn't scale as $1/r^2$; you must have confused it with the force which is the gradient (spatial derivative) of the potential energy. Second of all, the potential energy of a bound state – a pair or collection of particles/objects that attracted each other – is negative.
You may see it's the right sign because the total energy is conserved. The total energy is the kinetic one plus the potential one. When the oppositely charged particles get closer, they accelerate so the kinetic energy goes up. Because the total energy has to stay constant, the potential energy has to go down; and indeed, a negative number getting even more negative (the absolute value increases) is going down. Because it was zero at infinity, it must be negative when these particles are very close.
first of all the energy varies proportional to 1/r not 1/r^2. that's important.
let's take potential energy = 0 when separation distance r = infinity.
secondly, the energy is NEGATIVE for a positive and negative charge so that that it become a larger NEGATIVE NUMBER as they get closer. a larger negative number is actually LESS energy corresponding to a bound state.
I know that's slightly confusing but that's how you have to think about it.