Learn Swedish: Hon lär sig svenska. - She is learning Swedish.

The verb lära actually means "to teach". To say "to learn" in Swedish, you use the reflexive verb att lära sig, which literally means "to teach oneself". Since the subject is hon (she), we use the reflexive pronoun sig (herself).

If I wanted to say "I am learning Swedish", would I still use sig?

No, the reflexive pronoun has to match the subject. For the subject jag (I), the reflexive pronoun is mig, so you would say Jag lär mig svenska. The word sig is only used for third-person subjects like he, she, or they (han, hon, de).

Does lär mean "learns" or "is learning"?

It means both! Swedish doesn't have a separate "-ing" form (present continuous) like English does. The present tense verb lär translates to both "learns" and "is learning" depending on the context.

Why is svenska written with a lowercase letter?

Unlike in English, the names of languages and nationalities are never capitalized in Swedish unless they are the very first word of a sentence.