1- Imam Hussein, peace be upon him, left Medina with his family, to avoid being forced to pledge allegiance to Yazid, which would have led to a battle in Medina, because he would refuse the pledge. And the Imam did not want war at first, so he left Medina for Mecca, considering that it is forbidden there to bear arms and war for any reason.

2- It is reported that Abdullah bin Abbas told him to go to Yemen, for there are Shiites in your father, and he did not want to mention India, according to my reading.
As for the Imam, peace be upon him, it was reported that in order to establish the argument against the people in Karbala, he said to them: If you hated my coming after you invited me, then let me go to a safe place on earth, and there is no need to kill me.

3- Because he was not – apparently – going to war, but rather an answer to the call of the people of Kufa, who promised him to help him, so he did not need to take an army with him.
This is on the one hand. On the other hand, the Imam was not a ruler and had a regular army to lead him, so all he did, peace be upon him, was to invite people to go with him, and he told them that everyone who joined him would be martyred, because he was aware of what things would turn out to be, because the Prophet told so.