One of the primary themes is the necessity and power of love in guiding a person’s development. Harry is a gifted magician who must learn to develop his powers and use them appropriately. Underlying that gift and responsibility, however, is the love that his parents gave him.
Similarly, the relationship between nature and nurture is also a very important theme. Closely related to this is the importance of community, including both friendship and a larger support network.
Harry’s status as a wizard separates him from ordinary humans, whom the magicians call “Muggles.” Harry’s profound alienation in being raised by his Muggle relatives had a damaging effect on him, but the legacy of his parent’s love and sacrifice (symbolized by the scar) sustains him. When he is called to Hogwarts, a part of him is not surprised, as he knew he must belong elsewhere. Harry’s gifts were suppressed but not extinguished; it will be up to him how he develops that birthright.
When he arrives at Hogwarts, only some of the decisions regarding his life are within his power. The device of the sorting hat shows that Harry’s placement within a specific community, or house, is out of his hands. Here again, however, he must forge his own path in making a secure place within that community. While his talents are obvious, simply learning the magic lessons is not what Hogwarts is about—Harry shows his true kindness, generosity, and courage, which are some of the qualities that make him belong in Gryffindor.
I would say there are a great many themes to Harry Potter and the Philosopher's Stone. One of the major ones is discovery. For most of his life, up to this point, Harry is raised by his magic-hating Aunt and Uncle, in ignorance of the magical world and his own place in it. In the course of the first book, he will be introduced to an entire secret world of magic, a world in which he is viewed as a hero.
In addition, there are broad themes of good and evil introduced in this book that will be woven throughout the larger series that follows. Harry Potter's confrontation with Lord Voldemort at the end of the book will only be the first in a series of battles, with Voldemort being the main villain of the series as a whole.
Additionally, I would note that Harry Potter and the Sorcerer's Stone also addresses themes of privilege (which is represented by Harry's school rival, Draco Malfoy), and friendship, with Harry forging friendships with fellow students Hermione Granger and Ron Weasley. These friendships would prove critical in getting through the series of tests that were set up to guard the stone, and they would continue to prove critical throughout the books which follow.
A huge theme within Harry Potter and the Sorcerer's Stone is that of the power of choice and its impact on our wellbeing, on our lives, and on the lives of others. We are repeatedly given examples of difficult choices and their consequences or positive outcomes.
For example, the book begins with an examination of how choice impacts Harry's early life with the Dursleys. As a ten-year-old orphan living with his terrible Uncle Vernon, Aunt Petunia, and Cousin Dudley, Harry has very little agency; he is forced to live in a tiny cupboard beneath their stairs and is largely neglected. Even after the flood of letters addressed to Harry begin to arrive, Harry is not allowed to open them, and is, in fact, swept off to a small island to sequester him from the truth of his heritage and being: Harry is a wizard. Once imbued with this truth and provided with agency—the ability to choose—Harry's life is radically and irrevocably changed.
More and more instances of the consequences of choice arrive as the book unfurls: We learn that Harry's mother chose to sacrifice herself out of immense love when Voldemort hunted the Potters down, an act that resulted in her own death... but also in Harry being able to survive Voldemort's attack while being rendered mysteriously protected from his touch. We discover that Professor Quirrell chose to serve as a vessel for what is left of Voldemort's decrepit being, an act that results in Quirrell's slow unraveling and eventual death. We discover that Nicolas Flamel, the creator of the sorcerer's stone, makes the difficult decision to destroy it... an act that will eradicate the powers of immortality that the stone imbues and result in Flamel's death, but will otherwise serve the greater good of wizarding-kind.
We see less outwardly dramatic choices as well: When offered friendship by arrogant Draco Malfoy, Harry chooses to reject this relationship—despite its offer of instant power and social climbing in his new school environment—in favor of friendships with the far less popular Ron Weasley and Hermione Granger. Ron and Hermione end up becoming some of the most critical people in Harry's life and a source of continued support for him.
Harry chooses to break the rules during a broom-flying lesson in order to act honorably and rescue the Remembrall of his awkward peer, Neville Longbottom. While this is a risky choice, Harry winds up as the Seeker of the Quidditch team because of the boldness and skill he displays on the broom.
Ultimately, choice becomes an incessant presence within this book, as well as within the rest of the Harry Potter series. Perhaps what J.K. Rowling is getting at here is an important life lesson for us readers as well: We can either let life happen to us, or stand up and choose courageously... even if the choice is one that might result in pain or difficulties for ourselves. There is always power in seizing one's agency, in learning to deal with unfortunate circumstances, and in fighting on the behalf of the forces of good.
The main theme throughout the Harry Potter books is the triumph of good over evil. This is true in Harry Potter and the Sorcerer's Stone as well. However, applying this general theme of good vs. evil specifically to this novel, Harry must fight Voldemort's evil accomplice Quirrell to prevent Voldemort from getting the sorceror's stone and obtaining immortality.
I think another theme that is just as important to this particular book is the power of love. When Voldemort killed Harrry's parents, he was unable to kill Harry, and his attempt results in the scar on Harry's forehead. Dumblodore explains to Harry that although his parents are dead, their love for their son is always with him. Dumbledore explains to Harry that their love "is in your very skin".