When I visited the island, I flew into and out if Cágliari since it’s the closest airport to Rome. I rented a car and drove anti-clockwise around the island. I also took the ferry to Corsica and did the same there, The natural beauty of Sardinia, whether the mountains or the sea, is breathtaking. Realize that the island doesn’t have the major “wow” attractions found on the mainland. The city of Càgliari (Which I saved to the end of my visit) has a number of wonderful sites to visit including many churches, the Bastion and Castello, museums, Archeological sites and a nice botanical garden. There’s the Maddelena Archipelago in the northeast, the city of Alghero on the west coast and then further south the Roman ruins of Tharros on the Sinis peninsula and the quaint city of Oristano. Do not miss visiting the various Nuraghi throughout the island with the largest being the Su Nuraxi UNESCO World Heritage site near Burimini. In Alghero, walk the promenade of the bastion and take a boat to the Grotto de Nettuno.
A very typical Sardinian meal is Pasta alla Bottarga. Seafood is a fresh and delicious staple all over the island. In Càgliari, you might enjoy a traditional meal at Sa Domu Sarda, Via Sassari, 51 - make reservations. Wash it down with a glass of vermentino or cannanou.
If you can add a few days to your trip, take the car on the ferry and visit Corsica. I’d advise at least a day trip (with no car) from Santa Teresa Gallura to Bonifacio. It’s a beautiful harbor/marina with an imposing Citadel and interesting winding streets. I would recommend taking one of the sightseeing boats from the marina to sail out and see the white cliffs and the calanques.