in series resonance the voltage drops to that of the supply voltage across both the L and C, but you still have huge voltages across the L and C, which are out of phase, so they cancel out to give you the supply voltage across both of them
how do you move a massive amount of current? you need a large voltage, but voltage is measured from point to point, so you have to take into account exactly where you measure the voltage, and what is happening in between those points
impedance is lowest because the inductive reactance and capacitive reactance are equal and opposite - NOT because they are both zero... they are both large.