Learn Italian: Sai dove sono il nastro e le etichette? - Do you know where the tape and the labels are?

In Italian, there are two verbs for "to know". Sapere (used here as sai) is for knowing facts, information, or how to do something. Conoscere is used for being acquainted with people, places, or things. Since you are asking for a piece of information (where the items are located), sapere is the correct choice.

Why is the word order dove sono il nastro e le etichette instead of putting sono at the end?

In English, indirect questions put the verb at the end: "where the tape and labels are." In Italian, it is much more natural to put the verb right after the question word, even in the middle of a sentence. So it translates literally as "where are the tape and the labels."

Could I just say il nastro e etichette and drop the second article like we often do in English?

No, in Italian you generally need to include the definite article before every noun in a series. You have to specify il nastro (the tape) and le etichette (the labels), whereas in English we can easily skip repeating the word "the".

Why is the verb sono plural when the first item after it, il nastro, is singular?

The verb sono (are) has to agree with the entire subject. Even though il nastro is singular, it is joined by e (and) to le etichette. Together they make a plural subject (tape + labels), so you must use the plural verb.