Outdoor light fixture not working
My obligatory "get well-rounded knowledge" statement: the scenic way is to find an accessible-feeling book on the subject and read it through. Google is dangerous, that will leave you with swiss cheese knowledge. It only answers questions, and you need a well-rounded primer to know which questions to ask. 

Anyway you are on the right track.  Your measurements at the switch are valuable, and normal.  The reason the readings surprise you (especially re: neutral) is that switches don't use neutral. That wire is not neutral.  They connect (or don't) two wire functions: always-hot, and switched-hot.  Neutral is not in play on a switch, unless it's a modern smart switch.  

Voltage will always be zero (ish) across a switch with a broken lamp.  If the switch is on, the two terminals are both 120V.  If off, always-hot is 120V and switched-hot is unconnected/isolated becuse the lamp is broken, so no voltage between them.  If the lamp was working, it would pull the switched-hot line toward neutral, so you'd see 120V across the open (gapped) switch. 

Powering back up to check the voltage across the bare wires outside is a reasonable next step, just be careful.  120V across them would indicate a fixture fault, 0V would reveal a wiring fault.  



* most authorities think changing a light switch, receptacle or light fixture is too small a job to justify pulling a permit, so I interpret that as "too small to justify hiring an electrician", hence, handyman.