...Should something
>happen immediately? It does make noise when it runs - it's not silent
>like a thermoelectric fridge, right? I've never actually seen one
>running, so I'm not sure what to expect.

It is silent, except for the cooling fan.  The electricity and gas actually
heat a gas/fluid inside the cooling coils/pipes on the back.  Leave it
plugged in for an hour or so, keeping watch that nothing burns up.  If no
heat in back and/or cooling in front, then something is wrong.