The Kingdom of Sardinia is a misnomer. Admittedly a peculiar polity, it is the result of shrewd state-building by generations of rulers belonging to the House of Savoy, whose origins are to be found as Counts, later elevated to Dukes, in the eponymous region along the French-Italian border. Their aspirations to become a sovereign state, and be recognized with the status of Kings by the great powers of Europe, would lead them to govern the island of Sardinia. These ambitions would then play into the hands of activists who sought to unify Italy over the course of the 19th century. I've written an answer in two parts below.

The Kingdom of Sardinia

By the time of the War of Spanish succession in the early 1700s, the Savoyard state in northwestern Italy was composed of six contiguous personal unions accrued over several centuries. These six states were the the County of Nice; the Principality of Piedmont; the Marquisate of Saluzzo; the Duchy of Monferrat; the Marquisate of Susa; and the namesake Duchy of Savoy.

Although Piedmont was the most populous and prosperous terretory of the union, because of the peculiar history of urbano-centric Italian polities the oldest and most prestigious title was instead that tied to Savoy. However, this title was technically subordinate to the extinct Duchy of Burgundy, and thus only ever allowed the eponymous house to tread a fine line alternating between subordination to the Kingdom of France and its own distinct political line in Italian affairs. So becoming Kings of Italy, or even Kings of Sardinia, was never exactly an explicit objective. The Savoyards just wanted to be kings of something in order to definitively elevate themselves to the status of autonomous and sovereign monarchs. Success would come after the war of Spanish Succession, when the House of Savoy was fortunate enough to have backed the winning side: the Duchy of Savoy was elevated to the status of Kingdom by being awarded the island of Sicily. Very pleased with this, the newly crowned King Victor Amadeus II and the Savoyard court moved to Palermo. While their permanence lasted only about a year, the House of Savoy stayed just long enough to showcase all the most pompous royal proceedings to their court and foreign visitors alike (the Sicilians themselves, at least as much as the aristocracy was concerned, were bemused and satisfied that the Savoyards were much more interested in the ceremony of government than they were in the actual business of governing).

However (and here we can really begin to answer your question) when fighting between the great powers of Europe flared up a decade later, the Kingdom of Spain managed a rebound of sorts. The Peace of Utreict in 1720 handed Sicily to the Kingdom of Naples, a Spanish client-state and eternal claimants to the island (that is part of the reason why the Kingdom was successively styled "Kingdom of the Two Sicilies"). As compensation, the Savoyards were instead granted the island of Sardinia, a Spanish possession whose government also carried the title of kingdom. However, while Sardinia was much closer to the Savoyard's major port (the city of Nice) the exchange was seen as an overall downgrade: Sardinia was less densely populated, less urbanized, and less agriculturally productive than Sicily (the region's spectacular beaches, I imagine, were not sufficiently convincing as compensation). The court couldn't even decide if they should travel to the Sardinian capital of Cagliari for a coronation ceremony. In part the Savoyards were fuming about the loss of Sicily, but there was also a problem in that the title King of Sardinia had been extinct for so long no one was able to unearth instructions on how the ceremony was to be performed. This was a big deal, especially because the title King of Sicily had come with an elaborate ancient ceremony that the Savoyards had been happy to use in order to impress foreign attendees.

A ruler belonging to the House of Savoy wouldn't set foot in Sardinia for 78 years. The fateful day would only come in 1798, when Napoleon's invasion at the head of the French Revolutionary Armies forced the Savoyards to evacuate their capital in Turin and set up a rump-state in Cagliari.

The Savoyards did not recall their time in Cagliari fondly. While in the following decades the monarchy would have to grapple between their own local identity and notions of a unified Italian identity, there is no doubt that they did not consider themselves Sardinian. It was only with extreme reluctance that they abandoned the Piedmontese city of Turin, their capital at the head of Italy's great river Po, where they had ruled since the Italian Wars of the 16th Century. Over three centuries, senior leadership in the Savoyard administration had been exclusively the in the hands of luminaries and aristocrats from the populous cities and productive countryside in Piedmont, such that their society and culture would quickly eclipse the Savoyard's original identity hailing from the peaks and plateaus along the Franco-Italian border. The Piedmontese identity would even hold fast as the monarchy cynically issued titles, awards, patents, as well as commissions of regiments and ships under the more prestigious title of King of Sardinia.

Under different circumstances, Napoleon's reorganization of Italy might have turned the monarchy to embrace Sardinia and Sardinians. After all, the Piedmontese themselves did not miss Savoyard rule: after tumultuous years of war and famine, the new Napoleonic Empire not only offered stability, but for many also promised opportunity. The bourgeoisie of the small but elegant city of Turin had welcomed the chance to prove themselves in Napoleon's sprawling imperial bureaucracy. Even as Napoleon was crowned King of Italy in Milan, not a half mile away the new Palace of the Senate welcomed representatives from all of the new Italian Kingdom, fueling consensus and faith amongst a middle class that up until then had been growing in size, but had been stifled in influence. In Napoleon's massive army, ambitious officers who did not have the money or connections to obtain commissions found themselves promoted thanks to merit. Certainly, it must have been infuriating for the Savoyard monarchs to see themselves discarded so rapidly by their own subjects. However, they were also painfully, and probably bitterly, aware that the Island of Sardinia did not give them the resources to do anything about it. During long stays in Florence, Victor Amadeus III could only furiously appeal to foreign ambassadors to have his lands reinstated. His brother and successor, Charles Emmanuel IV, spent much of his time doing the same in Rome and Naples.

Cagliari today is a medium-sized Mediterranean shipping center. It is, notably, the station where much of Italy's crude oil is unloaded and refined before being delivered to gas stations and power plants on the Italian mainland. Although industry has fueled its modern growth, Cagliari is also a very green city, with large parks and avenues dotted with palm trees. The city slopes uphill from the port to the old town, so the sea never looks to be more than stone's throw away. The Royal Palace, where the Savoyard monarchs stayed during their visits, overlooks the sea and seaport at the edge of the promontory on which the old town is perched. It is altogether a pleasant place to live and work. However, at the time of the Napoleonic wars the Savoyard monarchy was very much unimpressed.

The artistic and intellectual life in Cagliari, while certainly much smaller than that in Turin, might nonetheless have fueled a pleasant courtly life. However, the monarchs seldom met foreign ambassadors in the capital of their rump state. Indeed, the generally loyal Sardinian aristocracy and bourgeoisie, much like the military contribution of Sardinian recruits who repulsed attempted French incursions and suppressed popular uprisings, went largely unappreciated. Upon Napoleon's exile to Elba, the King Victor Emmanuel I spared no time in relocating back to Turin. He had succeeded his brother, the aforementioned Charles Emmanuel, after the latter abdicated in a mixture of shame and disgust once he realized Napoleon was here to stay.

Victor Emmanuel, who referred Napoleon as "The Usurper," spared no time in dismantling Napoleon's provinces and recreating the old personal unions in his realm, each with their own courts and reliant on their respective local aristocracies. His reactionary rule went so far as to tear up Napoleonic roads and uproot the Napoleonic botanical garden. Napoleonic military officers who wished into re-enlist in the Piedmontese army were demoted. Exemplary is the anecdote belonging to the Napoleonic general Jean Maitheu Serras, who had been born in Savoy. Upon appearing for an audience at the Turinese court, he was promptly interrogated as to who he was by King Victor Emmanuel: "Serras, general of division, count of the kingdom, great eagle of the legion of honor!" answered the boisterous general, only to hear the king reply, "Fifteen years ago I had a Serras who was a sergeant in my guard. Farewell, my sergeant."

For their staunch anti-Napoleonic stance, the Congress of Vienna would award the Kingdom of Sardinia its seventh constituent state: the Republic of Genoa. The Savoyards now governed the whole of northwest Italy.

The role of the Kingdom of Sardinia, more descriptively but anachronistically also called Kingdom of Piedmont-Sardinia, follows below