Santiago de Chile Airport (SCL) is a hub for LATAM Airlines, so they operate direct flights to the region frequently, including some departing from Wellington. Flights departing from other cities in New Zealand will usually include a connection in Auckland. Another common option for direct flights to Santiago is Qantas . Malaysia Airlines also runs occasional direct flights from Auckland. Qantas often has the most generous luggage allowance on this route, but if you don't need much luggage, LATAM Airlines sometimes has cheaper fare options.
If you're visiting Santiago, Chile, as a tourist on a New Zealand passport, you won't need a visa as long as you are staying for no more than 90 days. You'll just need to have a passport with at least six months remaining after your trip. If you are a citizen of a country that does require a visa, you will generally need to apply for one through Chile's Ministry of Foreign Affairs. This can usually be done via Chile's online Consular Services website. You may be able to receive an e-visa or prior authorisation to pick up your visa when you arrive, depending on your nationality.
Car hire facilities are available at Santiago de Chile Airport, so you can pick up a vehicle when you arrive. You can find a number of international car hire providers, including Avis and Europcar, as well as the local Chilean Rent-a-Car on the first level, by Gate 2. You can use your New Zealand driving licence for up to three months, but you should also get an International Driving Permit beforehand. The cars are usually located either on-site or in a nearby car park that can be reached via a complimentary shuttle.