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Firstly, it's worth checking out the beautiful views from the top of the hill on which Anzi is situated. You won't find a much better vantage point in the area from which to see the villages around, the view all the way to the Murge (the rocky karst-like formations near Castelmezzano), nad the surrounding valleys and rivers.
While at the top of the hill, you can check out the early mediaeval church, which boasts recently restored mediaeval artworks, as well as a secret passage reportedly used by the Carbonari (the secret society which plotted Italian unification in the 19th century).
Then make your way down the hill into the village, through the winding streets, where you can check out the grounds of the old nunnery, and the beautiful abandoned palazzi (small mansions) of the old elite, including Anzi's more famous residents who participated in the Risorgimento.
Near the old municipal offices, you can see the presepe poliscenico, which is the fourth biggest in Europe, and which situates the story of Christ in settings from Anzi and surrounding villages, beautifully realised by local craftsmen.
It's worth then taking a walk down through the main piazza, the Piazza Dante, to the new restaurant which just opened in Anzi, the Osteria 'Il Cielo in una Stanza', which has brilliant local food, and where the owners can tell you all about local life and history.
Another option, if you want to get out into the local countryside, is to take everything you need for a picnic, and go to the nearby Diga della Camastra, a lake formed by damming the river Camastra, and an excellent place for a picnic on a hot summer's day.