Kagome was your typical middle school girl, until she was pulled through an old well and ended up in the Feudal Era, 500 years in the past! There she discovers a strange boy pinned to a tree, unconscious, and a whole world of trouble. The first bit of trouble is that she is found by villagers who think she is a spy from a neighboring warring state! The old priestess, Kaede, quickly ascertains that Kagome is innocent, but they still won't let the poor girl go home! Then she is attacked by a giant demon centipede who claims Kagome possesses the sacred Shikon no Tama, or Jewel of Four souls. Evidently the jewel is sought by many demons because of its great power. Running from the demon Kagome stumbles back to the tree with the boy. Amazingly her prescence seems to wake him, but he thinks she is someone else - the woman who pinned him to the tree 50 years before! After convincing him that she is not that woman Kagome is forced to make a choice. She can face the centipede alone and die, or free the boy, who can probably save her, but doesn't seem very nice himself. The old priestess seems to think the boy is more dangerous than the centipede, but Kagome, trapped against the tree by the centipede, decides to take a chance. Once freed the boy does destroy the centipede, but it turns out that he is after the jewel as well. He then attacks Kagome, but with the help of a beaded necklace infused with a spell (thanks to Kaede) Kagome is able to stop him. He can't hurt her now, because the necklace's spell allows her to incapacitate him with the word 'sit' at any time, but he refuses to go away.
The boy is a half demon named Inuyasha. 50 years before he had been in love with the priestess who protected the jewel, Kikyo. He was supposed to use the jewel to become human and be with her, but something happened and she became convinced that he had betrayed her, so she shot him with a sacred arrow - which should have killed him, but somehow only pinned him unconscious. Shortly thereafter she died as well, taking the jewel with her. Since the jewel was found in Kagome's body, and Kagome was able to remove the sacred arrow from Inuyasha, they decide that she must be the reincarnation of Kikyo. Side note - Kaede is actually the younger sister of Kikyo. Anyway, Inuyasha doesn't like Kagome, because she reminds him of Kikyo who he believes betrayed him, but he's sticking around hoping to get the jewel. He plans to use it to become a full demon (which is what he was going to do before he fell in love with Kikyo - that's how they met, he kept trying to steal the jewel from her) Unfortunately lots of other demons want the jewel as well. When it is taken Kaede sends Inuyasha and Kagome to go get it. They stop the demon, but in the process Kagome shatters the jewel and the shards fly all across Japan.
And thus begins the quest. Inuyasha and Kagome must set out to find the shards. And they must do it together for, as Kaede explains, only Kagome can see the shards when imbedded in a demon, and only Inuyasha has the strength to take them back. This doesn't sit well with either young person as Inuyasha still doesn't like Kagome, and Kagome has school back (er... forward) in her own time. But Inuyasha wants the jewel, and it was technically Kagome's fault so...
So they set off in search of the jewel, helping villages and picking up friends along the way. During the first season the pick up four additional members and two permanent enemies. The first enemy is Sesshoumaru, Inuyasha's full demon half brother who is searching for the legendary sword his father left behind. Unfortunately it appears the sword is destined for Inuyasha, which starts a long series of battles between the two brothers. The next main character we meet is Shippo, a young fox demon who has recently been orphaned. Inuyasha and Kagome save him and he stays with them. Next they run into a lecherous monk who is also collecting shards. He hopes to use them to draw out a powerful demon named Naraku who gave his family a generational curse that will one day destroy him if Naraku is not killed. The monk is named Miroku, and he reveals to Inuyasha that Naraku was the one who turned him and Kikyo against one another. Naraku attacked Kikyo disguised as Inuyasha which is why she came after him to kill him. Kagome convinces Inuyasha and Miroku to join forces.
And, of course, that was the introduction of the second villain - Naraku. Naraku is very powerful and very evil. He wants two things: the jewel, and the death of Inuyasha.
The last two members of the cast are picked up soon after. They are Sango, a young demon slayer, and Kirara, her partner, a fire cat demon. Sango's village was tricked and everyone except her was killed. The culprit... none other than Naraku! Naraku took control of Sango's younger brother, Kohaku, causing him to kill all the slayers. Kohaku is only kept alive by the jewel shard in his back. Sango is out for revenge, and since Inuyasha and Miroku are too, it only seems logical that she join them. So the six of them set off to find jewel shards and destroy Naraku.