I don't know how the Internet is changing the way I think because I don't know how I think. For that matter, I don't think we know very much about how anyone thinks. Most likely our current best theories will end up relegated to the dustbin as not only wrong but misleading. Consider, for example, our tendency to reduce human thought to a few distinct processes. We've been doing this for a long time: Plato divided the mind into three parts, as did Freud. Today, many psychologists divide the mind into two (as Plato observed, you need at least two parts to account for mental conflict, as in that between reason and emotion). These dual-systems views distinguish between automatic and unconscious intuitive processes and slower and deliberative cognitive ones. This is appealing, but it suffers from considerable anomalies. Deliberative, reflective cognition has long been the normative standard for complex decision-making — the subject of decision theory and microeconomics. Recent evidence, however, suggests that unconscious processes may actually be better at solving complex problems.
Based on a misunderstanding of its capacity, our attention to normative deliberative decision-making probably contributed to a lot of bad decision-making. As attention turns increasingly to these unconscious, automatic processes, it is unlikely that they can be pigeon-holed into a dual-systems view. Theoretical neuroscience offers an alternative model with 3 distinct systems, a Pavlovian, a Habit, and a Goal-Directed system, each capable of behavioral control. Arguably, this provides a better understanding of human decision-making — the habit system may guide us to our daily Starbucks fix (even if we no longer like it), while the Pavlovian system may cause us to choose a pastry once there despite our goal of losing weight. But this too likely severely under-estimates the number of systems that constitute thought. If a confederacy of systems constitute thought, is their number closer to 4 or 400? I don't think we have much basis today for answering one way or another.
Consider also the tendency to treat thought as a logic system. The canonical model of cognitive science views thought as a process involving mental representations and rules for manipulating those representations (a language of thought). These rules are typically thought of as a logic, which allows various inferences to be made and allows thought to be systematic (i.e., rational).
Despite more than a half-century of research on various logics (once constituting the entire field of non-monotonic logics), we still don't know even the broad outlines of such a logic. Even if we did know more about its form, it turns out that it would not apply to most thought processes. That is, most thought processes appear not to conform to cognitive science's canonical view of thought. Instead, much of thought appears to rest on parallel, associative principles - all those currently categorized as automatic, unconscious ones, including probably most of vision, memory, learning, problem-solving, and decision-making. Here, neural network research, theoretical neuroscience, and contemporary machine learning provide suggestive early steps regarding these processes, but remain rudimentary. The complex dynamics underlying non-propositional forms of thought remain an essential mystery.
We also know very little about how brain processes underlie thought. We do not understand the principles by which a single neuron integrates signals, nor even the 'code' it uses to encode information and to signal it to other neurons. We do not yet have the theoretical tools to understand how a billion of these cells interact to create complex thought. How such interactions create our inner mental life and give rise to the phenomenology of our experience (consciousness) remains, I think, as much of a fundamental mystery today as it did centuries ago.
Finally, there is a troubling epistemological problem: to know whether the Internet is changing how I think my introspection into my own thinking would have to be reliable. Too many clever psychology and brain imaging experiments have made me suspicious of my own introspection. In place of the Cartesian notion that our mind is transparent to introspection, it is very likely that numerous biases undermine the possibility of self-knowledge, making our thinking as impermeable to ourselves as it is to others.