What you're going to see if you do that Mike is exactly what you see now if you just look at any waveform before and after you put it through a filter. Do you really want to go through multiplying each sample by each coefficient and adding them together? IMO, it's going to be very tedious and not very enlightening.
But you can think about it this way - filters work by "remembering" the previous samples (the capacitor does this in the analog world) and using phase shift.
Consider that at a given sampling frequency there is a phase difference between adjacent samples based on the relationship between the frequency of the signal and the sampling frequency. At low frequencies the phase shift is very small - 0° at 0Hz (aka DC) and 180° at the Nyquist frequency (exactly 1/2 the sampling rate = exactly 2 samples per cycle = 180° phase difference).
So basically what is happening is you are phase shifting your original signal with the single sample delays, multiplying the signal that has been phase shifted by the appropriate coefficients, and then adding the result back into the original signal. This will either add or cancel
at a given frequency, and to varying degrees depending on the amount of phase shift at a given frequency and the value of the coefficient (including whether it's positive or negative).