The book starts out in Upstate New York where Percy is going to school at Yancy Academy. This is where he is attacked my Mrs. Dodds and eventually gets expelled and returns to his home in New York City.
When his mother realizes the danger he is in, she rushes him to the north shore of Long Island (pg 61) where Half-Blood Camp is located. This is where she supposedly dies, the Minotaur attacks, and where Percy finds out who his father is.
He returns to Manhatten to catch a bus and start his quest.
After they are attacked on the bus by the Furies, they look for food and find some at Aunty Em's Garden Gnome Emporium on the New Jersey riverbank. (pg 168) This is where they meet and kill Medusa and obtain her head.
They ride an Amtrak train to St. Louis, Missouri, (pg 201)where Annabeth wants to look at the architecture of the St. Louis Arch. Here they get attacked by the Chimera and Ecidna, and Percy jumps into the Mississippi River from the Arch.
From there, they travel to Denver (pg 219)where they meet Ares in a cafe, and he sends them on a quest to return his shield that he left in a water park.
They take an animal truck from Denver to Las Vegas, Nevada. (pg 257) This is where they are detained for five days at the Lotus Casino. This seriously cuts into the time of their quest and limits it to one day.
They then take a cab to Santa Monica, California,(pg 266) outside of Los Angeles where Percy meets Nersid, spirit of the seas, and receives the three pearls.
West Hollywood, California, a suburb of Los Angeles,(pg 274) is the location of Procrustus, the water bed salesman. Here they find out the location of DOA Records.
Valencia Blvd, Los Angeles, California, (pg 283)is the location of DOA Records. This is the entrance into the underworld, Hades Realm. Here they meet Charon, the riverboat ferryman, Cerberus, the three headed dog, and Hades, himself. Percy realizes he has Zeus's thunderbolt.
They return to Santa Monica Pier, California,(pg 319) and confront Ares for his role in the war making effort.
Flying back to Manhatten, (pg 336) Percy enters the Empire State Building.(pg. 336) The elevator takes him to the 600th floor, which is the entrance to Olympus, Zeus's realm. Percy returns Zeus's bolt.
Percy then returns to Half-Blood camp on the New Jersey shoreline where he finds out that Luke is the person who betrayed him.
Finally, he returns to New York City to live with his mother until the following summer.
Enotes is currently experienceing problems with the site. I cannot give you the exact reference notes, but Enotes has a site for The Lightning Thief and you will find a synopsis there that will help you.
The key places would be the casino in Las Vegas, Camp Half-Blood, the Underworld, maybe the classroom, which was where the plot begins.