Omega rental cars, good service, good price, free car seats for kids/baby, and they'll pick you up from the airport. Insurance is free but has a high excess, if you have travel insurance you may be covered for the rental car excess already.
Looks like you'll be travelling in peak holiday time, but try airbnb, or book a batch, or NZ holiday homes. Coopers Beach, and Matapouri are my favourite place to go with little kids. Many private camp sites have cabins, the Top 10 holiday parks are well worth a look, we absolutely loved our stay at the one in Taupo (clean/tidy with free bbqs, swimming pool, playground).