I have a gas stove in my new apartment (I've never used one before.) there is a wall mounted shut-off valve above the stove. Do I turn this (valve) on/off every time I cook, or can I leave it on all the time?

The wall-mounted valve is probably the normal gas supply valve. The knobs on the range turn the burners off and on.

You normally do not need to shut the gas supply valve off unless there is a problem with the stove or the stove is being disconnected for some reason.