Research basic drip watering systems. The essential design has not changed since drip systems were first developed. A main line, connected to a water source, brings water to where plants are located. Feeder lines from the mainline deliver the water to specific plants, via emitters. Essentially, water travels from the source to the plant via a pathway in between.
Find the water source for the drip system. Drip irrigation systems can tie in where the water supply comes to the house, or from a hose bib (faucet), or even from sprinkler heads off of a pre-installed system. Investigate all the points where water is available, and you're almost sure to find the main line for the drip system.
Follow the main line. Once you've found the system source, connect it and turn it on. Visually trace the line and walk along it as far as it's visible. If it's buried at some point, you can either dig it up and pull it gently out of the ground, or use the stethoscope to try to hear where the line goes. Verify what you hear with an occasional test hole to visually confirm the line location.
Follow the subsidiary lines. As you encounter feeders leading from the main line, you can trace those, too, which should lead straight to the individual plants the system serves. Since the system is on, water should be visible flowing to the plants.
If the main line is clearly delivering water, but some feeder lines are dry, there is likely a blockage in the thinner tube. Fortunately, drip irrigation system components are highly adaptable, fairly cheap and replaceable. Switch out any parts that prevent plants from receiving water. Lines can also be moved easily to accommodate landscaping changes. In this way, a "new" drip irrigation system can be developed from an old one without having to "reinvent the wheel."