Learn Italian: Chi cerca il mio cane? - Who is looking for my dog?

When chi (who) is the subject of the question, it acts just like lui or lei (he/she). That is why we use cerca, the third-person singular form of the verb cercare.

The English translation says "looking for", but where is the word for "for" in the Italian sentence?

The Italian verb cercare has the "for" built right into its meaning. It means "to look for" or "to search for", so you never need to add an extra word like per after it.

Why do we say il mio cane instead of just mio cane?

In Italian, you almost always put a definite article (like il or la) before a possessive word. Because cane (dog) is masculine and singular, we use il mio. The main exception to this rule is for singular family members (like mio padre), but pets do not get that exception!

How do we pronounce the c sounds in chi and cerca?

Chi is pronounced with a hard "k" sound, like "kee" in English, because the letters ch always make a "k" sound in Italian. However, the c in cerca is followed by an e, so it makes a soft "ch" sound, just like the "ch" in the English word "chair".