Cricket certainly, and rugby league also. Netball and hockey are also good sports for Australia at least. Basically all the sports of the British Empire educational system.
If only we knew someone who knew all about relative strengths of international teams in a myriad of team sports.
In terms of "Oceania" (and not just Australia and New Zealand), I think Rugby Union, Rugby League and Australian Rules football (all already mentioned) have the most Oceania representatives in the Top 20, say. Touch Football and Rugby Sevens (as derivatives of the above sports) can also be included.
As for hockey, outside of Australia and New Zealand, everyone else is pretty rubbish in Oceania - the same with basketball.
Cricket has Australia and New Zealand, while Papua New Guinea are probably in the Top 20. Vanuatu, Fiji are further down (but probably still in the Top 50).
In baseball and softball, Oceania has quite a few competitive middle-ranking teams like Guam, American Samoa, Palau, etc.