Learn Italian: Dove sono le magliette? Non le vedo. - Where are the t-shirts? I do not see them.

No! The first le in le magliette is the definite article meaning "the" for feminine plural nouns. The second le in Non le vedo is a direct object pronoun meaning "them" (referring back to the t-shirts). They happen to look identical, but they do different jobs.

Why do we use le for "them" here instead of li?

Direct object pronouns must match the gender and number of the noun they replace. Since la maglietta (the t-shirt) is feminine, its plural is also feminine. Therefore, we use the feminine plural pronoun le to say "them". If we were looking for masculine items, like i pantaloni (the pants), we would use li.

Why is it Non le vedo instead of Non vedo le?

In English, we put the object pronoun after the verb ("I do not see them"). In Italian, these short object pronouns almost always go before the verb. So, the literal word order is "Not them I see".

Does the non always go before the pronoun?

Yes. When a sentence is negative, non goes right at the beginning of the action, before the object pronoun. The order is always: non + pronoun + verb.