Short answers, pulled from the story.
The Kingdom of Sardinia was created on paper in 1297 by Pope Boniface VIII, who granted it as a fief to King James II of Aragon under a secret clause in the Treaty of Anagni. Its de facto existence began in 1324, when James II conquered Pisan territories on the island and asserted the title.
During the War of the Quadruple Alliance, Victor Amadeus II, Duke of Savoy, was compelled to cede Sicily to the Austrian Habsburgs and accept Sardinia in exchange. The transfer was formally ratified in the Treaty of The Hague on the 17th of February 1720.
The Kingdom of Sardinia was the direct legal predecessor of the Kingdom of Italy. On the 17th of March 1861, law no. 4671 of the Sardinian Parliament proclaimed the Kingdom of Italy, extending Sardinian institutions and laws across all annexed territories.
Although Cagliari was the de jure capital since 1324, the Savoyard rulers kept their government, ruling class, and centre of population entirely on the Italian mainland. Turin had been the capital of Savoy since the mid-16th century and served as the de facto seat of power. The Perfect Fusion of 1847 gave this arrangement official status by centralizing all governmental institutions in Turin.
In 1860, Giuseppe Garibaldi led a military campaign that toppled the Kingdom of the Two Sicilies in the name of the Kingdom of Sardinia. His forces conquered territory stretching from Abruzzo and Naples on the mainland to Messina and Palermo on Sicily before Cavour and Victor Emmanuel II moved to limit further advances.
James II of Aragon began the conquest in 1324, seizing Pisan territories in Cagliari and Gallura. The Judicate of Arborea resisted for nearly a century, at one point in 1368 reducing the Aragonese hold to just the port cities of Cagliari and Alghero. The contest ended in 1409 at the Battle of Sanluri, and the remaining Arborean territories were purchased for 100,000 gold florins in 1420.