Learn German: Liebe Anna, wie geht es dir?

Liebe translates to "Dear" in this context. Because Anna is a female name, the adjective takes the feminine ending -e. If you were writing to a man, you would use the masculine ending -er, such as Lieber Thomas.

Why do we use dir instead of du or dich?

The phrase wie geht es literally translates to "how goes it". In German, you express this condition as "how goes it to you". Therefore, you must use the dative (indirect object) pronoun dir ("to you") rather than the subject pronoun du or the direct object pronoun dich.

Could I just write wie geht's?

Yes, you can! Wie geht's is a very common casual contraction for wie geht es. However, because this sentence looks like the opening of a written letter or a nice email to a friend, it is very standard to write out the full words geht es.

How would this sentence change if I were writing to someone I address formally, like a teacher?