The flight durations you should expect on flights when making a trip from a city in New Zealand to Connecticut in the United States are going to be somewhat varied, based on where you plan on departing from in New Zealand and also which city you plan on arriving in when you land in Connecticut. For example, a flight from Auckland International Airport (AKL) is going to involve a travel time of about 29h 50min, whereas a trip from Christchurch Airport (CHC) is going to take around 32h 30min; both of these are examples of cheap flights to the United States. If you depart from Wellington Airport (WLG), you should expect a travel time that is about the same as the trip from Christchurch.
While on the one hand, it may be a little disappointing that you will not be able to find any direct flights to Connecticut from any city in New Zealand, the upside is that there are plenty of Connecticut flight deals you can find to make this trip. There is a three-stop flight that departs from Auckland International; it has layovers, at Sydney Kingsford Smith International Airport (SYD), Los Angeles International Airport (LAX) and Washington Dulles International Airport (IAD), before arriving at Hartford Bradley International Airport (BDL). There is also a two-stop flight from Christchurch, which has layovers at Sydney Kingsford Smith and Houston George Bush Intercontinental (IAH) and also arrives at Hartford Bradley International.
Because of its location in the northeastern United States, about halfway between New York City and Boston, there are plenty of road options to get you around the state. One choice is to hire a car and travel along the interstate highways you will find to take you to several points in the state. You can also take one of the many train options, either with the national train system called Amtrak or along some of the regional commuter lines that you will find throughout Connecticut. You can also make use of the different bus companies that offer services throughout the state, to get you between many of the large towns and cities in Connecticut.
Because the state of Connecticut has such a long and rich history, there are plenty of things for visitors to see and do when they visit here. The Mystic Seaport Museum is a detailed re-creation of a 19th-century maritime village, which makes for a great way to spend an afternoon exploring. The Mark Twain House & Museum is an interactive museum that offers very knowledgeable tour guides, who can give you both the history of the house and of its very famous residents. If learning about animals under the sea is more your thing, then make a stop at Mystic Aquarium, where you can see all manner of aquatic species up close. If the kids have made the trip with you to Connecticut, a very nice place to visit for the whole family is Lake Compounce, which is both an amusement park and a waterpark in one.
If you are a resident of New Zealand that has plans to visit Connecticut in the United States, you will need a passport that is valid. While you will not need a visa to travel to the United States as a tourist, you will need a travel authorisation. You can get one of these authorisations electronically through the website of the government of the United States; the cost is around $14 (NZ$ 21).