I am not sure the Internet has changed the way we think so much as the way we act. Information has become cheap, and spend more time on-line than in libraries, but there's been no biological evolution: human brains remain human brains, with a finite capacity for absorbing information and host of cognitive biases that impair our judgements. People have vastly more information at their disposal now, but it doesn't mean they know how to use that information wisely. Teenagers, for example, often gauge the reliability of a Website by how slick a site is, rather than on the nature of the site's sources.
My suggestion? Let us use the Internet as an impetus for completely rebooting our educational system, reorienting it from its current but antiquated 18th century emphasis on memorization — pointless in the age of Wikipedia — to a more modern emphasis on critical thinking skills, on metacognition and decision-making. Instead of teaching kids mere facts we should be teaching children how to reason, reflect, plan, investigate and evaluate.
If we can do that, then (and perhaps only then) we might truly change how people think.