I have an electrical issue with my 1978 Spider 124. The lights all work when the headlights are off - the directionals (front and back), the hazards, and the parking lights. However, when the headlights are on, the front directionals and hazards do not work, although the rear directionals and hazards still work. Any ideas?

From the sounds of it, something is shorting out when adding the additional load. Test your wires at the headlight switch.

Let us know what you find out if this isn't it.

Classic earth fault.

Trace all of the earth wires from your front lights and ensure they are clean and making a good contact.

FIAT are fond of using common earth points with multiple spade connectors that bolt to the bodywork. Unbolt these and clean them thoroughly, then coat them in grease (or preferably contact lubricant) and bolt them back up.
Do the same for any earth wires that bolt to the bodywork.