I've just arrived back from Niue today, so can give you some up-to-date information.
Market days are Tuesdays and Fridays, but I found them to be rather disappointing, not a lot being sold there.
The tourist office gives out a very useful pamphlet at the airport when you arrive. That lists the women's weaving groups, but it's best to check with the tourist office as to which ones are actually on, as for some of them it depends on whether they've got a village show day coming up. The group at Makini Hall in Alofi on Wednesdays is a reliable regular one apparently. I also went to the Peniamina group in Makefu, whose regular time is Tuesday mornings, but they stayed open well into the afternoon, and also opened on Thursday. Both these groups have a display room where they have items for sale. I also went to another group in Alofi on a Tuesday, which didn't have work for sale, but did have fresh cucumbers (useful if you're self-catering).
I was lucky enough to see a hair cutting ceremony, but this is a pretty rare event on an island with a population of 1400 people.
It's very easy to strike up a conversation with people. It's a bit like a country town, people wave at others and say hello, so if you see someone doing something interesting like hauling a traditional canoe ashore, just say hello, and start talking with them.