I just submitted my grades for my fall LING 101 course, and I'm busy preparing for my spring LING 101 course, so the question of how to get people to think scientifically about language has been much on my mind recently. I've found that one of the most useful entry-level questions is "How is human language difference from other forms of animal communication?" The media loves animal language stories, and so the uniqueness and complexity of human language is one of the first things I cover. One of the most important things about discussing such topics is not "Is human language unique?", but "What concrete properties of human language distinguish it from animal communication?" We're doing science, and so we want to point to specific criteria to distinguish the two; we want a theory of human language that predicts specific empirical facts. This is not how the general public usually thinks about language (or about anything; critical thinking is far removed from the natural pattern of human cognition).
Another topic I've always wanted to cover in more detail is speech perception. Often when I tell non-linguists that I work on how we perceive speech sounds and assign them to various categories, I get blank stares. Certainly before I was in linguistics I gave no thought to speech perception. When I lived in Italy in elementary school it was inconceivable to me that Italian speakers couldn't understand English; my first hypothesis (admittedly quickly discarded) was that when someone spoke English they simply heard nothing. We think of language as magic: direct communication from one mind to another. It takes a bit of work to transition into the type of thinking that evaluates the creation of sound by the human vocal tract and analyses how these sounds are transmitted as vibrations through the air, and then perceived by the human auditory and perceptual apparati (yes, I know that's not the proper Latinate plural). The question of how we distinguish a bilabial nasal from an alveolar nasal is not a natural one to ask, but it's an important question for linguists.
These are some of the basic concepts I'm planning to use in my 101 class next semester. If anyone has suggestions for other concepts useful for introducing people to the scientific study of language, I'd be glad to hear about them in the comments.