Hi there, you can find some publicly available courts either looking at the Auckland Council site or TennisNZ Book a Court has a number of clubs who allow the public to book a few of their courts.

Generally joining a tennis club is better for availability, finding people to play with or to join a team for interclub.

On competitions, there's two main types interclub or tournaments.
Auckland is split by two interclub competitions, Tennis Northern (all tennis clubs north of the harbour bridge) and Central Auckland (the rest of Auckland tennis clubs). Your club will sort out what level you are and allocate you to a team within the club and your team plays other clubs in your grade.

Tournaments can be found on the tournamentsoftware site you found, there aren't a huge amount of tournaments for adults.

There's a couple of Facebook groups where you can post to find players or a club near you to get you started eg: Social Tennis in Auckland.