At one level, the setting of The Lightning Thief is a world of demigods, monsters, and magic related to Greek mythology. The protagonist’s full name is “Perseus Jackson, Son of the Sea God,” and I would argue that this makes it immediately apparent that we aren’t dealing with any ordinary setting.

In the physical sense, however, this novel is set in various parts of the US, and at the beginning of the novel, Percy lives in New York and attends Yancy School in upstate New York, which is a school for kids described as troubled. Percy does not fit in at this school and describes his schoolmates as “rich brats playing at being bad boys.” We learn that this hasn’t been Percy’s “setting” for long, because it’s the sixth school he has been at in six years.

After Percy learns that he will not be invited back to Yancy the following year, we learn that the school has provided him with a scenic setting, with “views of the woods” from his dormitory as well as more distant views of the Hudson River. Other settings mentioned early in the novel include Queens, where Percy’s mother and stepfather live, and Montauk, where Percy and his mother went on holidays.

The next significant setting in the story is Camp Half-Blood, a camp for demigods and demigoddesses. Any discussion of this setting should include the fact that it is reminiscent of Ancient Greece, complete with an arena and amphitheater. This camp is full of magic and potential dangers, which Percy discovers when a hellhound is summoned into the camp. It is at Camp Half-Blood that Percy learns that he is “the son of the Sea God” and that there are some monsters who have an intense desire to see him dead.

Later, Percy’s journey takes him all around the US. Percy and his companions visit Gateway Arch in “the humidity of St. Louis” before heading to Denver, which Percy soon says he has “seen enough of.” They later spend time in a five-star Vegas hotel, which Percy, who has grown up in a relatively poor family, finds mind-blowing. Soon, however, he remembers that the purpose of their trip is to head to Los Angeles and “find the entrance to the Underworld.”

The setting of Percy’s adventure can be summed up in Chiron’s words:

America is now the heart of the flame. It is the great power of the West. And so Olympus is here. And we are here.

The action in this fine novel--isn't it fun to read!--moves from place to place, happening in many specific settings. The most important setting, though, is a contemporary world that is also mythic.

That is to say, the stories are set now, and all kinds of contemporary activity is referred to, but, at the same time, the gods, goddesses, and assorted creatures of classical Greek mythology are alive and driving events in the world. That means that the setting is a numinous version of our world (a spirit-filled or infused version). It also means that the setting is, if not a non-scientific version of our world, an other-than-scientific version of our world that is other-than-empirical. Two or more realities overlap at any time, both real for the people experiencing them.

Of course, if you want concrete settings, much of the first book is set in New York, places like Queens and Montauk Beach. And the most vivid setting of the first book is Camp Half-Blood, where young demi-gods live and are trained to be heroes.

Percy Jackson and the Olympians: The Lightning Thief, has many major and minor settings. Although the book involves authentic Greek gods, it takes place in modern day America.

As the book opens, Percy is attending a school for troubled boys in upstate New York. However, he actually lives with his mother and stepfather in New York City. Since New York City is now the present day site of the seat of power of the Greek Olympians, this setting plays a major role in the book.

Percy is sent to summer camp at Camp Half-Blood, also in New York but does not spend too long there before he is out on an adventure after finding that he is a demi-god, related to the ancient Olympians. 

From the camp, Percy sets out with friends on a  journey across America to Los Angeles, the site of the gates to the underworld, where he goes with other demi-gods to find Zeus' missing lightning bolt. After locating and capturing it, the setting of the story returns to New York. 

The Lightning Thief had 2 major settings and a few more minor ones throughout the text. It the opening and end of the book the story is based in New York City, where the story follows Percy through his school, museum, and surrounding neighborhood. Later, after he discovers his demigod nature, he is taken to Half-Blood camp to live with the other half-bloods. After that he embarks on his journey to find the lightning bolt spanning across much of North America, into Hell, then back to New York City for the final fight with the true Lightning Thief. 

If you need more specific details I suggest checking the chapters where these events took place and referencing them as well with your answer.

Hope that helps!