When I tried to invent a new rationality exercise, "Using Google" was one of the ideas. The exercise could be like this:

Give participants a list of many (non-mindkilling) questions they probably don't know, such as "How high is Eiffel's tower?". First let them comtemplate the questions somehow. (Perhaps tell them to guess the answers, make a probability estimates, or something. Just to let them feel that they don't know.) Then give them a computer and a short time limit (5 minutes), and tell them to answer as many questions as they can using Google. Then compare the answers in a group... and if there is some disagreement or suspicion, explore the topic more deeply (again, using Google).

The idea is to make people feel the contrast between how difficult the question felt, and how quickly and reliably it could be answered by using Google. This way they should be more likely to use Google in the future, when they are curious about something.

(Of course it is fair to note that this strategy does not necessarily work with mindkilling ideas, where people invest a lot of effort into creating and promoting web pages with incorrect information. Another good idea could be: in case of controversy, read the Wikipedia article and it's talk page.)