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Why do we say there is an overall negative charge on the intracellular side of the plasma membrane at rest, and an overall positive charge on the extracellular side when both potassium and sodium are positively charged ions, and they are in relatively equal amounts on either side? It seems to me that both sides should be positively charged, and I can't get my head around it.
The resting membrane potential is the voltage at which there is no net flow of ions across the membrane.
Every ion will tend to flow down it's concentration gradient (this is just a basic principle of physics/chemistry). So, without considering charge, for a typical cell, potassium will tend to flow out, sodium will tend to flow in, and other ions will also behave according to their concentration gradients.
The other crucial part of the equation is the relative permeability of the membrane to each ion. This comes mainly from the presence of ion channels. At rest, a typical cell has more permeability to potassium than sodium.
That means that if the driving force for sodium and potassium is the same (it's not identical at 0 mV but fairly close), more potassium will move than sodium.
If the definition of resting potential is the potential where net flow of charge is zero, and considering only sodium and potassium, then rest has to be where sodium flowing in equals potassium flowing out. This situation occurs when the inside of the cell is negatively charged compared to the outside: the membrane voltage "holds" potassium in and "pulls" some sodium in, such that their currents are equal and opposite despite the higher permeability to potassium.
Note that the voltages we are talking about in neurons, on the order of 10s of mV, are very small, and electrical forces are very strong. The concentrations of positive and negative ions are almost exactly identical on both sides of the membrane: only a few ions actually have to move to make a (biologically) large membrane voltage. What is actually important is the permeabilities: that's what makes action potentials and other forms of neuronal signaling feasible.