Australia, New Zealand

A few ways to approach this.
Simply put in shore excursions at (fill in blank) port into a search engine like Google. Sites such as Viator and get your Guide will appear with possibilities for that port. You can buy through those sites or find the name of the tour operator in the listing and book direct.

Consider
Is there free cancellation in case of last minute changes in itinerary?
Time in port: if irregular operations on the tour (traffic jam, etc) will there still be a good cushion of time to return to the ship?
Tender port: usually guests will ship sponsored excursions have priority for the first few tenders in a port. Does your private excursion give you enough time prior to the tour to make it off the ship?
Distance from ship and modes of transport. The further from the ship and the more types of transport you take the risk factors increase.

Private excursions are always a possibility, just use good sense when booking.

In addition to the Google searches, have you looked at the Ports of Call forum on Cruise Critic?
https://boards.cruisecritic.com/forum/48-australia-amp-new-zealand/
The availability of independent tour operators or advisability of going DIY will vary with each port. As Carol mentioned, the further away from the port you need to travel, fhe more careful you need to be in your choice if independent touring. If you are late getting back to the ship, the ship will not wait for you.

Another website that can provide info is WhatsInPort.

Thank you, helpful suggesions!

I highly recommend Mondumo Tours. Colin Mars is a Rick Steve's guide for Scotland but he does his own tours of New Zealand and Australia. My wife and our travel friends thought this was on of the best tours we had ever been on.

Mondumo Tours --> https://mondumo.com/