After a bathroom leak, who do I call first — my insurer or a plumber?

You should call a plumber first so any further damage can be prevented. Once the leak has been stopped and fixed, you can call your insurance company to make a claim so they can assess the damage. Your insurance company may deny the claim if they find that it was a slow leak that caused damage over a substantial period of time, rather than a sudden incident that would have been difficult to prevent.