My immediate thought there was the flasher relay. Pull this out and either test it outside the car or swap it for a known-good one.
If that is not the fault, then as you rightly point out, you need to make sense of the wiring diagram... The key thing here is the wire colours. Most cars have two-colour wiring, with a main colour and a trace colour (i.e. the little stripe down one side). If you can identify that you ought to be able to identify where each wire enters and exits the loom. However, it is usually easier to identify the circuit components and work from them.
I don't have access to a Chevy wiring diagram, but I would imagine it to work as follows:
battery -> ignition switch -> fuse box -> flasher relay -> signal switch -> bulbs -> ground.
(this is slightly different if hazard warning lights are fitted, I don't know if you use them over there? or are they a European thing? )
Obviously you know the battery and ignition are working, so I'd start by removing the fuse and relay and checking as follows:
- 12v at fuse box 'live'
- ground at bulb earth.
- Relay power and ground
- continuity between fuse output and relay input
- no short between 4 and battery -ve
- switch operation (disconnect it if you can)
- repeat 4 and 5 for relay->switch and switch->lamps.
But I'd still suspect the relay first...
(p.s. 1993 isn't old, it's not even 20 yet!)