Learn Italian: Noi siamo gli amici di tuo suocero. - We are your father-in-law's friends.

You are exactly right that Italian normally uses a "the" before possessives, but there is a major exception: singular family members! Because suocero (father-in-law) is a single family member, we drop the article and just say tuo suocero instead of il tuo suocero.

Why is it gli amici and not i amici?

Amici is a masculine plural noun starting with a vowel. In Italian, the masculine plural word for "the" is usually i, but it changes to gli before a vowel to make the pronunciation smoother.

Why do we use di to say "father-in-law's"?

Italian doesn't have an apostrophe-s ('s) trick to show possession. Instead of saying "your father-in-law's friends," you have to flip the structure around and say "the friends of your father-in-law" (gli amici di tuo suocero). The word di translates to "of" here.

Do I have to say Noi at the beginning, or can I just start with Siamo?

You can completely drop Noi! In Italian, the verb form siamo already means "we are," so adding Noi is optional. You usually only include it if you want to add special emphasis, as if saying "We are the friends..."