How can I tell if it's as simple as the neutral safety switch and not
a full blown clutch?
I would normally expect a neutral safety switch on a manual to prevent you from starting the engine, not getting into gear once the engine is running. This is the behavior that my car exhibits: if you don't have your clutch pedal pushed down, the car won't start.
So, if the car was already running, a bad switch doesn't sound like the culprit.
How did the car behave when you tried to engage first? Did the gears grind? Did the stick move at all?
How did the other gears sound? With a usable clutch mechanism, you should be able to engage top gear at a dead stop (although you won't be able to move, most likely).
If all the gears sound bad, I would be concerned about the clutch. If only first is bad, I would be worried about a gear or a synchro.
Final question: did this occur suddenly? Were you driving fine at one stop light and things were broken at the next light? If so, think back: was there a distinctive sound or symptom of the failure? A mechanical failure of this sort would likely have been very noticeable. An electrical fault could have been much more subtle.
Responding to the comments
When you say that you couldn't get out of gear, what do you mean? Does the stick move at all? Can you push the stick towards one of the gear gates? Do the gears grind when you do? Or is there silence?
If you can't move the stick at all, I would be concerned about the shift linkage.