A safe, though roundabout and inefficient, way to do the on/off sensing is to put the lamp on a switched outlet, such as a power strip. That power strip will also have a 5 V (assuming you have a 5 V Arduino, otherwise 3 V) power supply (like a phone charger) also plugged in. You turn the lamp on via the power strip, which also turns on the 5 V power supply, which is detected by the Arduino. Of course that doesn't directly tell you power draw but that should be a steady value. If you get a device such as a "kill-o-watt" you can just measure the current of the light once.
You could also detect lamp on/off using a photoresistor but you'll need to make sure that it wouldn't be falsely triggered by external light sources, such as sun coming though the window, while the lamp is off.