Lessons learned for me personally (so far) from the accident with the Japanese reactors:
=> Do your homework in advance. Nuclear physics is a complex field. I don't intend to learn every possible nuclear reaction formula under the sun. But I do want to learn enough to translate data - like an anomalous high radiation measurement - into the important question: What does this MEAN? Exactly how dangerous is it, and under what circumstances? What us the best course of action?
I've done my share of learning and re-learning some basic physics, and this process takes TIME. Even more so if your background isn't that of the natural sciences.
The same applies to other kinds of disaster (tsunami, chemical power plant etc). But I dare say that those things are far simpler to learn about than the basic nuclear physics and how nuclear contamination will affect you in the short and long term. (By the way, the answer to that is "not very much" for all but the absolutely worst catastrophic scenarios in proximity to the source).
=> Don't expect the authorities to give out factual info until they are forced to do so. Even with the best intentions, they may be overwhelmed, things are happening too fast and some bureaucrats at some level will inevitably fall victim to "wishful thinking" and dictate that the public should only be told the most possible positive interpretation of events.