Traditional Tongan dancing can be witnessed at the Tonga National Center on Tuesday and Thursday nights, at Fafa Island Resort when sufficient guests are present, at the Kahana Lagoon Restaurant on the Fanga'uta Lagoon Friday, and at the Good Samaritan Inn on Friday nights.
Package tours with dinner and transfers are available to all of the above. Reserve all cultural shows the day before as they're often fully booked.
The privately run 'Atenisi Institute, in the western part of the city, began as an evening high school in 1963 but it now offers university-level courses right up to PhDs. It has an international reputation for its classical school of thought—a bastion of critical mindedness and creative thinking in a conformist country.
If you're an "expert" on anything, Professor Futa Helu may invite you to lecture the students during the school year (February-October). Write in advance or make an appointment soon after you arrive. There's no pay, but someone might end up buying you lunch.
The traditional graduation ceremonies in November are not to be missed if you happen to be there at the time. (The Institute receives no government or church funding, and is in urgent need of overseas assistance.)