Learn Spanish (Latin America): No trabajo el sábado y el domingo.

Yes, el usually means "the". But in Spanish, to say you do something on a specific day of the week, you use the definite article (el for one day, los for plural days) instead of a word like "on". So, el sábado literally translates to "the Saturday", but it means "on Saturday".

Do I need to use el before both days, or could I just say el sábado y domingo?

In Spanish, it is the standard rule to repeat the article (el) before each noun in a list. Saying el sábado y el domingo is the grammatically correct and natural way to say "on Saturday and Sunday".

Why aren't sábado and domingo capitalized?

Unlike in English, days of the week (as well as months and languages) are not capitalized in Spanish unless they are the very first word of a sentence.

Where is the word for "I" in No trabajo?

In Spanish, the ending of the verb tells you who is doing the action. The -o at the end of trabajo (from the verb trabajar, meaning "to work") already means "I work." Because of this, you can drop the word for "I" (yo) completely.