Learn Spanish (Latin America): Te prometo un regalo.

In Spanish, object pronouns like te (which means to you here) usually go immediately before the conjugated verb. So instead of saying I promise you, Spanish structures it as to you I promise (Te prometo).

Is Te a direct or indirect object here?

It is an indirect object. The direct object is the thing being promised (un regalo), and the indirect object is who receives that gift (te). Verbs of communication and transfer, like prometer (to promise), dar (to give), and decir (to tell), almost always use these indirect object pronouns to show who is receiving the action.

Do I need to include yo (I) in this sentence?

You do not have to! The verb prometo already tells us that I am the one promising because of its -o ending. You could say Yo te prometo un regalo if you wanted to emphasize that I am the one promising, but dropping the yo is much more natural and common.

Could I attach te to the end of the verb, like prometote?