The electrons don't move very quickly either. Like, a sluggish one millimeter per second is more current density than most metal conductors can handle without melting. Thankfully, there's lots of mobile electrons carrying charge (coulombs) so that's a lot of current. "Electricity" only travels near the speed of light because voltage is like a force sending waves through the electric field (simplified). And it's instantaneous current (amps = ~~joules~~ edit: coulombs / second) times voltage (electric field potential difference in volts = joules per coulomb) that delivers power.

Simplifying to a single harmonic (pure 50Hz/60Hz sine voltage source and a passive, linear RLC load), you need not only multiply the voltage's and current's effective amplitude (that gets you apparent power in VA, voltamps) but also their power factor or cos φ (the cosine of phase beetween them) to get power in W (joules per second). If the cosine is one, it's a purely resistive (R) load (like a heater) with a phase difference of 0°. If the PF is zero, it's a purely reactive (L/C) load (a freewheeling synchronous motor is much like that) with a current phase of ∓90° and no power is consumed overall. If the cosine is negative, power is actually being generated by the device you're measuring (for instance, old elevators and escalators with synchronous motors are actually delivering power into mains when enough people are travelling down).