Since you mentioned wanting to learn a few things, here's some that might be instructive.
Feel free to ignore this if you find it confusing;
you can definitely troubleshoot without it. I only add it because you asked.
We find that your set of symptoms—bypass and LED working, but no effect—is usually related to soldering problems. Occasionally a misplaced component can also be the culprit, hence the need for larger photos, but usually it's a bad solder joint. Here's why that is true:
As explained in the link duhvoodooman shared, the signal path for bypass is very simple: in at the input jack, wired to the IN pad, down through a trace in the PCB to the eyelet marked 4, into the footswitch and back to the PCB at the pad marked 8, back up through the PCB trace to the OUT pad, and on to the output jack. When you engage the footswitch, the signal is
instead routed into the PCB components that make up the effect circuit before going on to the output path. At the same time, the LED connection is made, and the light comes on. (You can read about mechanical bypass switching in detail in
this excellent thread by Stephen.)
What often happens when you have a poor solder joint is that the audio signal hits an electronic "wall" at that point in the circuit and can go no further. You mentioned your multimeter. It is useful for testing continuity, but it won't help with your audio signal. (Continuity is the absence of resistance, so, for example, one end of a piece of wire should have continuity with the other end.) You can use the multimeter to make sure of a good
connection between two spots in the circuit, but you can't test to see if audio is going through. For that you'd need a signal tester, aka audio probe. You could buy or make an audio probe to determine exactly where you are losing the signal, but it would take less time to simply reflow all your solder joints, especially with an effect this simple. And it would be very likely to solve the problem.
Hope that's helpful and not too wordy. Let us know if you have more questions!