To get Residence, you have to apply under the SKilled Migrant Category, meeting all the conditions. And yes, the first stage of this is putting in an EOI.
http://www.immigration.govt.nz/migra...migrant/apply/ There is a lengthy verification process, so you may well not be granted Residence until a year or more after you first put in your EOI.
When you have held Residence for at least two years, you can apply for Permanent Residence, if you meet the conditions.
http://www.immigration.govt.nz/migra...quirements.htm And no, your time on other visas before you held Residence does not count towards this two years. It is only the main applicant who has to fulfil the conditions, so if you get PR, so will your husband and children, even if they are elsewhere.
After you gain PR, you have the right to come and go from NZ whenever you like.