First a word about formatting, Harry Potter and Cursed Child: Parts One and Two is not a novel, nor is it written as a novel. It is a stage play and is written as such. Based on an original new story by Harry Potter author J.K. Rowling, with John Tiffany & Jack Thorne, with the play by Jack Thorne. Some folks might find it problematic to read in such a format, but it didn't bother me, and after a while, honestly, I didn't even notice it.
The story is that of Albus Severus Potter, the son of Harry Potter and Ginny Weasley, and Scorpius Malfoy, the son of Draco Malfoy. Neither fit in well at Hogwarts, neither are good at making friends, and so they become each other's friends - the sons of two enemies of their youth. While Albus struggles with living up to his father's reputation as the savior of all wizardom, Scorpius must deal with rumors that he is the son of Voldermort. A bond is formed between the boys, similar to that of Harry, Ron, and Hermione in the original books.
For another view, check out Sarah Hawkins Miduski's thoughts on the book at Biff Bam Pop! right here.