Learn Turkish: Adresi arıyoruz. - We are looking for the address.

This is the accusative case suffix, which marks the definite direct object of a verb. Because we are looking for a specific address ("the" address), adres gets the suffix -i to become adresi.

Where is the word for "for" in Adresi arıyoruz?

In English, we say "looking for", but the Turkish verb aramak already means "to look for" (or "to search"). You do not need a separate word for "for". The thing you are looking for acts directly as the object of the verb.

The dictionary says the verb is aramak, so the root is ara-. Why is it spelled arıyoruz and not arayoruz?

When a verb root ends in a vowel (like the a in ara-) and you add the present continuous suffix -ıyor, that final vowel changes to fit vowel harmony. The a in ara- narrows to an ı, making it arıyoruz.

Doesn't aramak also mean "to call" on the phone?

Yes, aramak is a very common verb that means both "to look for" and "to call". You have to use context to figure out which meaning is intended. Since you cannot dial an address on a phone, here it clearly means "to look for".