What material could I use to transition between teaching my students Shakespeare and the Augustan Age?

Given the additional details, I'd think about setting up the transition by establishing some sort of narrative arc for the course as a whole on the first day. One possibility would be to talk about genre in light of literacy levels. Poems and drama can appeal to a primarily illiterate audience because they are normally performed (lyric poetry was originally accompanied by music), while the proliferation of prose depended on the rise of vernacular literacy. For this sort of approach, you might choose some songs such as "Fine Knacks for Ladies" or "Drink to me only with thine eyes" that are widely available on YouTube in your Elizabethan section and then move on to talking about Defoe's prose appealing to middle class Dissenters.
For a technical school, you might use some of Defoe's The Complete English Tradesman, and talk about the rise of the novel as part of training the ambitious tradesman in the manners needed to succeed in retail business, linking the rise of the novel to conduct and politeness literature such as etiquette and letter-writing manuals. Swift would then fit as a satirist teaching correct behavior by criticizing improprieties. Students in a technical college might respond well to this very pragmatic understanding of literature as offering behavioral models, as they are dealing with this issue in their own lives. Especially for first-generation college students, part of succeeding in careers is learning the unspoken codes of certain types of middle class society (clothing, body language, accent, etc.) and you could ask your students to compare their own situation to that of the 18th century rising middle classes.

There are several possibilities. I think you might want to use a brief lecture as a bridge and talk about the Puritans, the Restoration, and the move toward tolerance after the traumas of the religious wars of the intervening period. 
A few works might help with this. A Letter Concerning Toleration by John Locke might be a good example of the shift in values towards reason and restraint. On a more literary level, one might use Pope's "Essay on Criticism" as a way of showing how literary tastes changed. His "Essay on Man" contrasts well with Milton, but is less important if you skipped over Milton. 
Some of how you handle the transition depends on what you are doing in your class. If you are focusing on poetry, this was a great age for satire, and Pope would be a good focus. If you are thinking about the rise of the novel, Watts is a bit dated but a good starting point for the conversation. Essays by Addison or Steele might also be a good way of establishing the new ideals of taste. 
To answer this more fully, I'd need a sense of the pedagogical context. The strategies that might work for a seminar in drama wouldn't fit an introductory lecture course.