Learn Turkish: Takımımız çok çalışkan. - Our team is very hardworking.

It comes from the word takım (team). To say "our", we use the possessive suffix -(I)mIz. Since takım ends in a consonant and its last vowel is an undotted ı, 4-way vowel harmony tells us to use the exact suffix -ımız. So, takım + -ımız = takımımız (our team).

Since a team is made up of many people, why don't we use the plural suffix before adding "our"?

Even though a team has many members, we are only talking about one team. So, takım remains singular. We just attach the "our" suffix to it to get takımımız. If you wanted to say "our teams" (more than one team), you would add the plural suffix first to get takımlarımız.

Where is the word for "is" in this sentence?

In Turkish simple present statements, there is often no actual word for "to be" (am/is/are) for the third person (he/she/it). Simply putting the subject takımımız next to the description çok çalışkan is enough to say "Our team is very hardworking."

Is the word çalışkan related to the verb çalışmak (to work)?

Exactly! The verb çalışmak means "to work" or "to study". By adding the suffix -kan, it turns into an adjective describing someone who does that action a lot. So, çalışkan means "hardworking" or "studious".