To know what is on in Cagliari and neighbourhood, check the site Cagliari Turismo, the tourist portal of the City of Cagliari. Or check the site of Sardegna Turismo.
Of particular interest are the archeological sites of Barumini and Nora. The first for its Nuragic Culture. The second as one of the best kept Punic Quarters to date, whose anfitheatre is sometimes used for concerts, theatrical events and poetry recitals. To check for archeological sites, click here.
Sardinia is also famous for its jazz festivals. And Cagliari often hosts concerts worth listening to.