New Zealand Law
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The US citizenship will not affect your ownership of your house and the mortgage. It doesn't matter what country you are a citizen of, to own a residential property and get mortgages. There are no restrictions on you. So you can still buy other property if you get US citizenship. Once you marry, you can apply under the usual procedure. But it's not important, unless you actually want to become a citizen for other reasons, although you will need to get permanent residency to stay and work there.
Foreign citizens who want to live permanently in the United States must first obtain an immigrant visa. This is the first step to becoming a lawful permanent resident.Immigrating to the United States is an important and complex decision. In this section, you will learn about who may immigrate to the United States, the different types of immigrant visas, the required forms, and the steps in the immigrant visa process. Because most immigrants receive visas in the family or employment based visa categories, they are a key focus of this section. To be eligible to apply for an immigrant visa, a foreign citizen must be sponsored by a U.S. citizen relative, U.S. lawful permanent resident, or a prospective employer, with a few exceptions, explained below. The sponsor begins the immigration process by filing a petition on the foreign citizen’s behalf with U.S. Citizenship and Immigration Services (USCIS).
A U.S. citizen can file an immigrant visa petition for:
To begin the immigration process, your sponsoring family member must file an I-130 Petition for Alien Relative with U.S. Citizenship and Immigration Services (USCIS).
So the first step is to get this form from the US consulate in Auckland and then get the first stage of the permit. Once you have been there for long enough you can then move to applying for citizenship
You can travel to the US visa free at first under the ESTA program. You can start the application process for a partnership visa once you are married, either in New Zealand or over there. The ESTA is here https://www.estavisaonline.org/