One of the most important ways to "think like an economist" that students get taught is that "accounting costs are not the same as economic costs". When most people think of the cost of a certain action, they think of the accounting costs. Economists think more broadly about opportunity costs, and general equilibrium effects which may even feed back into accounting cost calculations.
I saw a good example in the Washington Post yesterday that demonstrates that most journalists don't think like economists when it comes to costs. It reads: "Obama jobs plan could cost $300 billion". That, of course, is the accounting cost. Even if you disagree with me on the impact of stimulus, if you think like an economist this shouldn't be the primary cost you consider when thinking about the stimulus. Indeed if you disagree with me on the stimulus you're likely to assume that the economic cost is even higher than this $300 billion accounting cost.