Although I'm rather unprepared to talk about actual fighting and military aspect of the conflict, I can speak to the reasoning behind the Kingdom of Sardinia's participation in the Crimean War and the quid pro quo it was meant to catalyze.

After the disastrous war between the Kingdom of Sardinia-Piedmont ^1 and the Austrian Empire in 1848, the Piedmontese saw the outbreak of the Crimean War as a chance to get in the good graces of the great powers of Europe and gain favors that could be exchanged for aid which would again allow them to declare war on Austria. The moment was particularly ripe to exchange favors to the detriment of the Austrians, as the Empire had peeved the French and English by flip-flopping on possible participation during the first year of the war. ^2 Plus, the the Piedmontese were the next-best ally anyway, as they had already been planning a Crimean military expedition following talks with the Austrians who had asked them to accompany them in a hypothetical intervention (the logic being that by tying down a not insignificant portion of the Piedmontese army, in a joint campaign no less, King Victor Emmanuel II and his sneaky Prime Minister Cavour wouldn't be able declare war on a distracted Austria).

So when the English and French asked the Piedmontese to participate, most of the government (including Cavour and the King) were very quick to say yes. In the treaty of alliance between France, England, and the Kingdom of Sardinia, the wily Cavour snuck in a clause guaranteeing Piedmontese territorial integrity and defense for he duration of the war, which would form the basis of the anti-Austrian Franco-Piedmontese alliance signed in 1859 (the English were less keen on returning favor, conssidering Austria as a pivotal counterweight to Russia).

1. I'm pretty bad with calling this bizzarre polity a consistent name. Generally, I will refer to the state by its official name: the Kingdom of Sardinia. However, I will use "Piedmontese" as its demonym.

2. Understandably, the Austrians were subject to a frantic Anglo-French courtship: if the Austrian Empire were to have opened up a wide second front against Russia, the course of the war would very probably have been much different.