The Titan's Curse
Rachel first meets Percy at Hoover Dam. She startles him, and he swings a sword at her (which passes through her due to her being a mortal), which results in her thinking that he is crazy.
The Battle of the Labyrinth
Percy has thoughts about Rachel, wondering who she is and fascinated by the fact that she could see through the mist despite being a mortal. Rachel meets Percy again at Goode High School and seems to be more interested in him. She gives him her number on his hand as Percy runs away from the school after setting it on fire; and Percy memorizes it, surprising himself. He also realizes that he needs Rachel to guide him, Annabeth, Tyson, and Grover through the Labyrinth. He defends her whenever Annabeth makes rude and sarcastic comments about her, which was probably due to the fact that she was jealous of Percy and Rachel being so comfortable together.
The Last Olympian
Rachel kisses Percy before he leaves for a battle, but he is confused since he also has strong feelings for Annabeth. Percy never really reaches a decision about what he feels for Rachel. Prior to her becoming an Oracle, Rachel had feelings for Percy and she kisses him for the last time just before she goes into the big house (which offended Annabeth), but the "relationship" ended after Rachel decided that she wanted to be the Oracle. They still remain good friends. After Annabeth found out about this, Rachel and Annabeth had a better relationship with each other and the two became friends as well.
Perachel is rivaled by Percabeth, but Percabeth was made canon in the end as being the Oracle, Rachel wasn't allowed to date Percy. Perachel is also notably a disliked ship, as evidenced by this. Perachel is treated badly by a lot of the fandom, but certain users are trying to bring it back in popularity, under the knowledge it will never be canon.