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!