I will list all of their meetings between Inu no Taisho's grave and his saving her from Mukotsu with details to help you more:
1. After Miroku and Shippou join Inuyasha and Kagome, Sesshoumaru makes a deal with Naraku and gets a human arm with a jewel shard in it to be able to yield Tessaiga. He takes the sword from Iuayasha and Kagome shoots an arrow at it. Sesshoumaru attacks her and fights some more with Inuyasha. Then Kagome shoots another arrow at his left arm, breaking his armor. He catches the third arrow and then attacks Kagome with one of the sword attacks. More fighting with Inuyasha who gets serriously hurt and end up breaking off Sesshoumaru's borrowed arm. Sesshoumaru leaves after that.
2. Sesshoumaru is looking for Toutousai to forge him another sword, but he has run away to find Inuyasha. Sesshoumaru flies in on Ah-Un and starts fighting with Inuyasha. Nobody gets hurt and Toutousai makes a distraction and the group runs away on his cow. Sango has also joined in by this time. Sesshoumaru gets a dragon arm and hunts down Toutousai and the tachi. More fighting and Inuyasha unleashes the Wind Scar on Sesshoumaru, injuring him pretty badly. He is only saved by Tenseiga's barrier and is transported away. Kagome did not get involved in any way. This is when Sesshoumaru meets Rin and saves her.
3. After Tessaiga is broken and reforged by Toutousai, Inuyasha fights and kills Kaijimbo, an evil sword smith that made Tokijin. Sesshoumaru is investigating a change in Inuyasha's scent when Tessaiga broke and yields Tokijin. Inuyasha is heving trouble with the new Tessaiga and losses control over his youkai blood and is about to attack when Toutousai makes a distraction again and takes everyone away. Sesshoumaru might actually have been a little afraid of Inuyasha at that time. Kagome again did not participate.
4. When attacked by a moth youkai, Inuyasha again losses control of his youkai blood and kills a horde of thieves. Sesshoumaru had found out from Bokusenou (an old tree youkai friend of his father) that Inuyasha is in danger of losing his soul after to many transformations. Sesshoumaru tries to keep Inuyasha in check. They fight but Sesshoumaru doesn't actually harm him, but throws him away with Tokijin. Kagome yells at Sesshoumaru to stop fighting and stay away from Inuyasha who is passed out on the ground. She calls him 'stupid'. He tells her to use Tessaiga to bring Inuyasha back to normal. When questioned by Miroku as to why he didn't kill Inuyasha he says that there was no point in his current state but that he will do it in time. He walks away after that.
5. Naraku kidnaps Rin and at his castle he tries to absorb Sesshoumaru's body inside of his own, but that doesn't work and the brothers fight Naraku together. When Inuyasha breaks naraku's barrier and injures him, Sesshoumaru actually lets Naraku get away in order to save Rin, who is held somewhere else by Kohaku. The rest of Inuyasha's group is already there trying to fight of a lot of demons. Kohaku takes Rin away and tries to kill her because he's under Naraku's control. Sesshoumaru and Inuaysha try to stop him and Kagome tells Sesshoumaru that Rin is fine, just fainted. Kohaku tries to fight Sesshoumaru and he chokes the boy. Kagome tells Sesshoumaru to stop and Kohaku runs away. Kagome thanks Sesshoumaru for not killing the boy. He tells her and Inuyasha that he doesn't want to play by Naraku's rules and then leaves with Rin.
That's all until the affair with Mukotsu. Sorry that this post is so long, but I thought the details will help. Also, if you want to watch the anime for more reference try
www.animecrazy.net/inuyasha-anime/ It has online episodes both in Japanese with English subtitle and in English. The episodes are 18-19, 34-35, 45, 52 and 81.
