I can't say the internet is changing or has changed the way I think. It has however changed the way I post-process what I think in several ways. This has pros and cons.
Pro: The most obvious change is that I share my thoughts with many more people than before. This has frequently resulted in very interesting feedback, opened my eyes to issues I neglected or points of view I wasn't previously aware of. This is one of the prime reasons I'm writing this blog.
Con: On the flipside, while writing down my thoughts I'll typically do some Google searches and come across previous articles on related topics. This likely affects my own opinion, and I'm not sure this is entirely a good thing. And, needless to say, some of the feedback I got has merely taught me that the world is full with ignorant, hostile, and simply crazy people. Knowledge I could I have lived without.
Pro: Clearly, the internet provides a vast amount of easily accessible resources. 15 years ago reading a journal article required going to the library, erring around in search of the right aisle, not finding the ladder, waiting half an hour till the guy with the ladder is done erring around, then realizing that the very volume you're looking for is missing, etc etc. Nowadays, it's a click on a link (unless your acrobat reader has crashed again). If it would take much more than that I probably wouldn't read articles in any other field than physics, so the internet has certainly broadened my horizon.
Con: On the flipside, this is a hard time for perfectionists. If you're trying to read everything available on a topic, you'll never finish anything. So when I'm writing I'm constantly trying to balance the amount of input with the expected benefit of the output, meaning I have to find the right point to stop reading. This typically will leave me with a bad consciousness. All these people, they had something to say too, and lazy me didn't read it.
Generally, the internet has changed what knowledge I regard relevant, and I suspect this is a quite widespread change. Now that you can fast and easily look up a lot of facts, learning them by heart is totally yesterday. Like, who cares if I can't name all presidents of the USA? What's the capital of Qatar again and when was the transistor invented? The problem is though that if you don't have any factual knowledge you won't even know what to look for. So I just hope that modern school education carefully selects what knowledge is really necessary to pipe into children's brains.
Another clearly noticeable change is the obsession with the present that the internet has brought upon us. A week from now, this post will have wandered down the "recent" list and nobody wil recall what I wrote. Maybe it's my European genes that object on the idea that only the Now really exists, but if we don't honor the past we'll just repeat our mistakes. Why does Google return recent entries first? What is it that makes Americans believe what's newer is necessarily better?
Maggie Jackson in her book "Distracted" warns, backed up by research studies, that this "Now-Culture" severely affects the capability of children (meanwhile teenagers) to sustain attention. We're now seeing the first generation grow up that was born with the Internet. If there's any major impact on human cognitive processes caused by the overflow of information we're faced with and the amount of tasks we have to simultaneously deal with then this development can become an obstacle to progress. Something to have an eye on. There's mistakes you only make once.
The other development that I've been writing about (eg in my post "The spirits that we called") is that naive mishandling of information can be a danger for democracy. This point was recently also made very well by Lawrence Krauss in his SciAm essay "War Is Peace: Can Science Fight Media Disinformation?"
"English novelist George Orwell was remarkably prescient about many things, and one of the most disturbing aspects of his masterpiece 1984 involved the blatant perversion of objective reality, using constant repetition of propaganda by a militaristic government in control of all the media.
Centrally coordinated and fully effective reinvention of reality has not yet come about in the U.S. (even though a White House aide in the past administration came chillingly close when he said to a New York Times reporter, “We’re an empire now, and when we act, we create our own reality”). I am concerned, however that something equally pernicious, at least to the free exercise of democracy, has."
So, for now my conclusion is that while I doubt the internet has yet actually changed thought processes, it has certainly affected what we think about. And in the long run, the latter is going to affect the former.