Learn Polish: Serio, nie mam czasu. - Seriously, I don't have time.

Yes, all three can mean seriously or really, but they have slightly different registers. Serio is highly colloquial and extremely common in casual conversation. Naprawdę is the most standard word for really, while poważnie translates closer to seriously in the sense of a serious situation or not joking. In everyday speech, starting a sentence with Serio is a very natural way to emphasize that you are telling the truth.

If I wanted to say "I don't have time for this," how would I expand this sentence?

You would use the preposition na followed by the accusative case, making it Serio, nie mam czasu na to. Even though czasu is in the genitive case because of the negative verb, the preposition na still independently dictates the accusative case for the pronoun that follows it.

Just as a refresher, why is it czasu instead of czas?

This is the genitive of negation. The affirmative phrase is mam czas, where czas is the direct object in the accusative case. When you negate a Polish verb that normally takes the accusative, the object must switch to the genitive case. Since czas is a masculine inanimate noun, its genitive ending here is -u.

Would a native speaker ever include the pronoun ja in this sentence?