At least for my kids, the series that is hands down the most like Harry Potter is the Percy (aka Perseus) Jackson series by ex-English teacher Rick Riordan. As in the Harry Potter series, the main protagonist discovers something about himself and ends up in a magical place.
Combining Greek myths with modern America (Mt Olympus is now on top of the Empire State Building) and learning difficulties with special powers (dyslexia helps with reading ancient Greek), there is a lot to love about this series, which seems to captivate kids anywhere from age 8 to age 14. If Harry Potter is set in Hogwarts, the Percy Jackson series is set at Camp Halfblood, a summer camp in Long Island, and it is from there that the adventures start out. If Harry has Hermione and Ron, Percy has Annabeth and Grover.
The book to start with in the series is The Lightning Thief (or you can take the plunge and buy the Percy Jackson boxset: it’s hard to go wrong). Rick Riordan has also written a number of other series you can move on to, which are explained by 13-year-old Alex (my son) here.
From our article Books like Harry Potter