Any Turin bucket list starts with the Musei Reali (Royal Museums), a group of sites that will take you through different eras in the city’s history.
Included in the same ticket and connected by one route, they consist of the infamous Palazzo Reale (Royal Palace), the first and most significant of the royal residences of the House of Savoy in the region of Piedmont (don't judge it by its modest facade), the Royal Armory, the art collection of Galleria Sabauda, the Museum of Antiquities with the remains of the Roman theatre, and the Chapel of the Holy Shroud.
Admission is €15, but it's free to visit with the Torino+Piemonte Card.