Tidus' relationship with Lulu is a continuously-changing one. This is mostly due to Lulu's changing view of him. She went from being annoyed and hesitant toward Tidus at the beginning of the game to respecting him near the end. I think Tidus eventually won her over by sticking with the party and completing their journey with them, ultimately defeating Sin and Yu Yevon.
Lulu's annoyance with Tidus is clearly evident when she met him in Besaid. She didn't like him because he was careless in the Besaid Cloister of Trials, entering the Chamber even though he knew he wasn't supposed to. After later learning how deeply she cared for Yuna, it's easy to see that Lulu was irritated that Yuna might have to pay the price for Tidus' foolish actions.
Lulu's annoyance with him wasn't helped by his limited knowledge of Spira, either. I don't think she believed his lie about Sin's toxin affecting his memory. Despite that, Lulu became somewhat of an informant to Tidus, explaining unfamiliar terms and ways of their world, eventually explaining things without Tidus even asking. For much of the game, that's pretty much the extent of their relationship.
Tidus overheard a couple conversations between Lulu and Wakka about him. The first was in Besaid and was about Wakka's deceased brother Chappu, Lulu's fiance. Lulu and Wakka apparently thought Tidus looked a lot like Chappu (What were they smoking?), and Tidus reminded Wakka a lot of him. Lulu told Wakka that Tidus was not Chappu, and that he shouldn't have brought Tidus to the village. I think Wakka getting so attached to Tidus because of his resemblance to Chappu worried Lulu. She saw that Tidus distracted Wakka, and she wanted Wakka focused on Yuna's pilgrimage.
I think Tidus realized that Chappu was a sensitive subject between Wakka and Lulu. While he felt comfortable enough with Wakka to ask about Chappu, I think he knew better than to ask Lulu. I also think knowing how Lulu felt about Tidus journeying with them may have been why Tidus never intentionally tried to become close with Lulu.
The second conversation Tidus overheard was about him becoming Yuna's guardian. Yuna (correctly) believed that Tidus was the son of one of her father's guardians, Sir Jecht. Lulu was trying to persuade Wakka to ask Tidus to become Yuna's guardian. I thought this was a weird request to come from Lulu. Tidus annoyed her, after all. I think her loyalty to Yuna and her desire to make Yuna happy made her put aside what she might have wanted herself. Maybe she had started to warm up to Tidus, though. She quickly began to respect him as a member of the party and was content to fight beside him.
Lulu eventually stopped getting annoyed with Tidus. The two began to talk about all kinds of things, like Zanarkand, Yevon, marriage, and even Chappu.
Even though she was warming up to him, Lulu's relationship with Tidus complicated when Tidus began to fall for Yuna. Lulu knew the terrible reality of Yuna's pilgrimage and didn't want to see Yuna hurt more than she had to be. Yuna would die after defeating Sin, but that was something that Yuna had accepted. Lulu knew that a relationship with Tidus would make things harder for Yuna at the end, and Yuna's well-being was Lulu's primary concern.
If she is to marry, I would want her to marry for love. But... If Yuna said she wanted to marry the one she loves, I would have to object.
Lulu knew that Yuna was falling for Tidus and warned Tidus not to fall in love with Yuna. When he said it was too late, she told him he would need to keep his feelings inside.
Tidus had become close to the two people Lulu cared most about, so I knew it was only a matter of time before she would accept him. I even think she became fond of him near the end of the game.