So first, let's sort out who actually wrote what. From what I can surmise, the story is a collaborative work by Rowling, Thorne, and Tiffany. But the screenplay itself was written by Jack Thorne.
So what to do. It's not technically a novel, is it? And because it's not, my expectations are slightly different, perhaps even lower. So where to start? Maybe at the beginning.
Mr. and Mrs. Dursley, of number 4, Privet Drive, were proud to say that they were perfectly normal, thank you very much. Okay, maybe we'll fast forward a bit and start at the beginning of this tale, as much as I'd like to recap all the books. Harry Potter and the Cursed Child takes place about 20 years after book 7 ends. Harry is married to Ginny and together they have 3 kids. The story focuses on Harry's middle child, Albus Severus, who is having trouble adjusting to wizarding life at Hogwarts. Of course, we learn of larger issues in the wizarding world as well. For after what seemed to be a fairly peaceful time in the wizarding world, signs of change are on the horizon. The story evolves into a complex tapestry, weaving this way and that, and even turning back on itself. This is because The Cursed Child involves time travel, which isn't my favorite story-telling device. But I was able to overlook that.
The good? Rowling was involved in writing the story. And she is great at good versus evil. The tale is strong, dark, and continues the themes of her previous books. You get Harry, Ron, Hermione, Draco, and a lot of other characters we know and love (although some just have quick cameos). She is even able to bring back some beloved characters that have not made it (at least in the world of the living) to this point, and she is able to do it organically and in a way that makes sense to the story.
The bad? It's a screenplay, so you lose a lot of the narrative and background you get with a novel. This is a huge pitfall, perhaps bordering on the ugly. Transitions that are better made in theater with music and lighting changes are not fully realized by reading the script alone. So going from scene to scene is a bit of a bumpy road. I also feel the story depends almost to a fault on the fact that you are familiar with the Potter books, While this is a bit of a given, even with the books, Rowling is able to remind us gently of certain things, either with a memory, a conversation, or a description. But with The Cursed Child, there are time and format limitations that result in certain things, like Albus' first few years at Hogwarts, for instance, being handled like a Fantasyland ride. (If you're not familiar with this story-telling device courtesy of Disneyland, it goes something like this: Snow White's Scary Adventures - quaint cottage. Snow White. Dwarves! Spooky woods. Pretty witch. A mine! Ugly witch. Scary woods. Witch at door. Dwarves!..and witch on mountain. Happy music. The end).
If you approach the Cursed Child as a Harry Potter book 8, you'll be disappointed. It will remind you of how wonderful Rowling's books were to read and made you wonder how much more information we could have obtained if this were a fully realized novel. If you approach it for what it is, a special peek into the theater production, you'll find that the show will be amazing with a lot of special effects and a fun story. I'd really only recommend the screenplay to people who don't expect to see the play but want to know the next canonical story in the Potterverse. For everyone else, I'd suggest seeing the play instead.