Suspected mold and water damage inside bathroom walls: where do you start?

It’s going to be a major pain no matter how we look at it. But what’s the best way to tackle it? What kind of contractor or inspector do we call?

A bath and shower unit was installed in place of the old bathtub about 25 years ago. The wall of the unit is now bulging, suggesting water buildup (or worse) on the other side. There are also small scattered black patches on the ceiling, which was originally painted with oil-based paint and later with latex, and in the caulking.

We’ve never dealt with anything like this, but in an old house I suppose it’s inevitable. How should we proceed?