Tidus doesn't really have much of a relationship with Kimahri. For much of the game, Kimahri doesn't give Tidus the time of day. Like Lulu though, I think Kimahri is eventually won over by Tidus. I think that has a lot to do with Yuna.
Kimahri first met Tidus in the Besaid Cloister of Trials. Tidus was there unsanctioned, and I think that rubbed Kimahri the wrong way. Before they left Besaid for Kilika, Kimahri attacked Tidus, forcing him to fight against him. The fight continued until Wakka intervened and broke it up. Tidus asked Wakka and Lulu what was up with Kimahri, naturally wondering what he did to be attacked. The two didn't know, and Yuna interjected that they didn't understand Kimahri either, because he hardly spoke. She did make one thing clear about Kimahri, though. Kimahri protected Yuna since she was a child.
It's never said why Kimahri attacked Tidus, but knowing how dedicated Kimahri was to Yuna, and that he took his job as guardian very seriously, willing to die for Yuna if he had to, I drew the conclusion that Kimahri just didn't think Tidus was good for Yuna. It's easy to see why Kimahri wouldn't want Tidus around her. Tidus entered the Besaid Cloister of Trials even though he knew he wasn't supposed to. That kind of wreckless behavior would only come to hurt Yuna at some point.
Although Kimahri doesn't try to attack Tidus again, he grunts or ignores Tidus when Tidus tries to speak to him during much of the game. Tidus continues talking to him, though, and eventually Kimahri speaks back.
One of the biggest scenes that stands out to me with Tidus and Kimahri was right after Sin killed most of the Crusaders. Yuna tried to remain upbeat, and Tidus commented on her cheerful nature. Kimahri turned around and said that in dark times, Yuna had to shine bright. He told Tidus that Yuna worries when they worry, so they shouldn't frown. He said he would try to remain cheerful, too. Jokingly, Tidus said, "Smile! Let me see," and Kimahri smiled (rather roughly, haha).
From that point on, Kimahri speaks more to Tidus in the game, and the two have a mutual respect for each other. I can't be certain what brought about Kimahri's change of heart about Tidus, but he probably saw that Tidus wasn't going to harm Yuna, and that he actually made Yuna happy. Yuna was the most important person to both of them, and I think that could have helped them appreciate each other more.