When I read Harry Potter and the Cursed Child I had mixed feelings. Because the book is based off a play script, it felt as if it was a completely unique style. Over time I got use to the new style, and to a extent preferred it over a normal book. The only draw back I felt with the reading style was that it couldn't capture as much character emotion as a typical book could. This lack of emotion I think would be the biggest draw back that any reader would experience.
When reading I felt as if the main theme could be seen in the main character - Albus. This theme is self identity, which I could definitely relate to. Throughout the book there are obvious uses of media, peers, family relations, and other influences that put the Albus into a confused state of mind. This leads Albus onto a course where tampering with time and dragging others down a path to a Tophet like place becomes his reality. If a person told me that self indentity wasn't something they struggled with, I would tell them they were lying. In this day and age, media delineates what a person should do, and be like in their everyday life.
Through this though, I think readers can find a secondary theme. This theme can be seen in Albus's best friend, Scorpius. This friend gave Albus a person to confide in, and similarly someone who supported him for who he truly is. Scorpius silently showed readers what it means to be a true loyal friend. I think that most readers would look at Scorpius as what a true friend should be, and maybe even change their lives based on his influence. To a extent a reader may even look at Scorpius and realize that they haven't been a good friend. They may even realize that they need to change their life signficantly before anyone could consider them a good friend.
The third theme that can be seen is the exact opposite of the second. Betrayel. In this book we see the main antagonist, Delphi, betray the two boys for her own gain. Delphi originally came up to the boys and acted as if she actually wanted to be their friends. Then later, she essentially black mails them into helping her do her own dirty work. I think most readers can look at Delphi, and find somene in their life that they've offended and betrayed.