If you are planning to stay in the new place for less than 3 months then you don't have to do anything. As per RTO rules it is considered as a visit. If you are going to stay for more than 3 months and less than one year then you have to pay a nominal Tax amount in the new state. But RTO officers will not accept tax for other state vehicle. They will ask for change of address and re-registration.

As per rules if you are planning to stay in the new place for more than one year you have to apply for change of address and re-registration. To apply for the change of address you have to get a NOC (No Objection Certificate) from the home state RTO where the vehicle is registered.

Insurance is valid all over India. But always inform the insurance company if you change address.