Sardinia has it’s own culinary traditions, but not a uniform one throughout the island. While there are some delicacies you can try anywhere, very often what you eat depends on the area you’re visiting. Some have more meat and others more fish-based dishes, and there are very local kinds of pasta and other specialties.
If you’re particularly desperate to try something different and local, and you have the courage, go for “Casu Marzu”, (aka rotten cheese). Just make sure you don’t eat the maggots that work so hard to give this cheese its distinctive taste and texture — by the way, they’re still alive.