In short, Zetsu's only goal, reason to exist and plan was to make sure Obito revived Madara. Black Zetsu is Madara's will, in Zetsu's body.
If you need explaining in scans, Madara was talking to Zetsu telepathically. When Chõjuro cut down the Black Zetsu, he was unable to move. Madara was disappointed and facepalmed because of this fail and ordered him to go to Obito in order to help him and make sure that Madara was revived. He merged with the ground, and was waiting to attack Obito all the time. He was a backup plan in Madara's sleeve.
The first scan was about Zetsu trying to get Obito's Rinnegan, which originally belonged to Madara. Since Obito was to die anyway, it could've been harmful if he died doing a suicide attack at Madara. Minato stopped Zetsu, who then later fused with Obito due to the similar cells in him within. He stated to be a loyal servant of Madara, and will make any use of Obito in anyway he can in Madara's favor.