Rikku is the first member of the main party that Tidus actually meets. She's a young Al Bhed girl who comes to Tidus' aid when he is attacked by a monster at the Baaj Temple Ruins. She wanted to help him, and although her actions are questionable, she seems to have good intentions. Tidus is knocked out and drug on board Rikku's ship. There, he is forced to sleep on the deck, and is told he can stay if he is useful. He helps the Al Bhed look for treasure.
While with the Al Bhed, Rikku takes care of Tidus. She gets him food, and helps explain things, so that Tidus can figure out what has happened to him. Rikku is the one who told Tidus about Sin's toxin, something that affects people when they get too close to Sin, and causes them to lose all of their memories. Tidus used that as an excuse when he didn't know something basic, as he felt the truth was too hard to believe.
Tidus told Rikku about Zanarkand, and she told him it was a holy place, a city destroyed a thousand years ago. I don't think Rikku really believed that Tidus was from Zanarkand, but she wanted to help him. She arranged for Tidus to stay with them until they got to Luca, a big city where surely he would find someone who recognized him. She didn't get to take Tidus to Luca, though. Sin attacked their ship, and Tidus was thrown overboard. Tidus passed out, and awoke on the shore of Besaid.
While in Besaid, Tidus dreams of Rikku. Yuna and Rikku both want to go to Zanarkand with Tidus, and he feels he has to choose which one to take. His father comes into the dream and taunts him, and then Rikku and Yuna help Tidus conquer the fear of his father, by yelling that he hates him.
Tidus looked for Rikku in Luca, hoping that some Al Bhed would know her. He wanted to thank her for saving his life, and wanted to know if she was alright. He didn't find her, though.
Tidus eventually meets Rikku again at the Moonflow. Tidus and Wakka unknowingly fought her when she captured Yuna with some machina. Though her methods were questionable, she was trying to save Yuna's life. Tidus finds Rikku laid out on the ground, and approaches her, happy to have found his old friend.
Rikku! You're Rikku! Hey! You're okay! How you been?
Rikku and Yuna discover they are cousins, and Rikku becomes one of Yuna's guardians, finally joining the party.
Tidus and Rikku are close. They were able to bond, both being outsiders of the group. They both seemed to really care about Yuna, and they jump to each other's defense when the group doesn't see eye to eye with them. Tidus displays his loyalty and friendship to Rikku by keeping the fact that she is an Al Bhed away from Wakka, who hates the Al Bhed.
Rikku is a possible love interest for Tidus in Final Fantasy X. You can select options throughout the game to have Tidus flirt with Rikku. It never gets anywhere, though, as Tidus is clearly supposed to be with Yuna. I think that Tidus and Rikku have more of a brother-sister type of relationship, and it works for them. They're pretty similar as characters, excited and full of life.