In answering the question, I would propose that Heinlein's work follows in close form to the themes inherent in the science fiction genre. One such theme is the idea that science fiction works are not really about other worlds as much as they are about our own. When we analyze science fiction worlds, we are analyzing our own state of being. Adam's "Hitchhiker's Guide to the Galaxy" is one of many science fiction works that does such a thing. I suspect that Heinlein, writing in the 60s, is playing on a similar theme here. The trick is identifying the concept being analyzed and applying it to both Earth and the setting of the science fiction novel. I will give the earthly concept, and then phrase some questions that you can use to complete the analysis. One social issue within the book is the issue of discrimination, both racial and gender. Heinlein is examining on earth what happens when people are locked into racially and gender stratified roles. How does he show this in the book? Another social issue brought up is the idea of childcare. How does he critique the "modern" version of childcare where parents pursue careers, sometimes at the expense of connecting with their children? The last social issue is one of dreams. What does it mean to dream? Heinlein is writing at a time when dreams are very prevalent in American society. The 1960s (book written in 1963) was a decade dominated by dreams and envisioning what can be as opposed to what is. How is this seen in the book? The opening lines might give us a clue and the fulfillment or deferring of dreams is a social issue we still wrestle with in contemporary society. How does the society in Heinlein's world embrace the notion of the dreams of youth? How do we embrace this idea in contemporary society?
These are fairly decent starting points for analyzing social issues that Heinlein comments on in both his book and in our world.