Learn Dutch: Geef het mij, alsjeblieft.

When a sentence has two object pronouns, the direct object (het, meaning it) comes before the indirect object (mij, meaning to me). If the direct object were a regular noun, the order would flip, for example: Geef mij het boek (Give me the book).

Could I say Geef het me instead of Geef het mij?

Yes, Geef het me is very common in everyday speech. Using the stressed pronoun mij instead of the unstressed me adds a bit of emphasis, as if you are saying give it to me specifically, rather than to someone else.

Why doesn't the verb Geef have a subject like jij next to it?

This is an imperative sentence, which means it is a command. Just like in English, you drop the subject when telling someone to do something. You simply use the stem of the verb geven.

Can I use alstublieft at the end of this sentence instead?

You could, but it would mix informal and formal language. The command Geef without a subject is informal, which perfectly matches the informal alsjeblieft. A fully formal version would add the formal pronoun u and use the formal please: Geef u het mij, alstublieft.