Given the complexity you’ll need to get a good meter to test things out.
While a simple DVM will allow you to test the circuits (power and wire continuity) it won’t tell you much from the tubes perspective. Here’s a useful meter [https://www.fluke.com/en-us/product/electrical-testing/basic-testers/fluke-1000flt|Fluke 1000FLT Electric Fluorescent Light Tester] while it’s not cheap! You might be able to rent one locally.
You might consider just re-wire the fixture to LED’s. I’ve re-wired a few ceiling fixtures using the [https://www2.meethue.com/en-us/p/hue-white-and-color-ambiance-lightstrip-plus-na-ext/719025548|Philip’s HUE Lightstrip] which allows even colorful mood lighting over my dining rm table! The tricky part was finding a place to hide the AC transformer.