Learn Polish: Zrobię to jutro, daję słowo. - I will do it tomorrow, I give you my word.

Zrobię is the perfective future tense. It focuses on the result, meaning you promise to complete the task. If you used the imperfective będę robić, it would mean 'I will be doing it' or 'I will spend time doing it', which doesn't sound like a solid promise to get it finished.

Is daję słowo a common idiom in Polish?

Yes, it is used exactly like the English phrase 'I give my word' to make a promise. Notice that in Polish, you don't need to say moje (my) or ci (to you) here. Just daję słowo is perfectly natural and complete.

Why is the second verb daję in the present tense while the first verb zrobię is in the future tense?

You use the future tense zrobię because the action of doing the task happens tomorrow. However, you use the present tense daję because the act of promising (giving your word) is happening right now as you speak. Using the future tense dam (I will give) would sound like you are going to promise tomorrow.

I know the sentence means 'I will do it', but where is the word for 'I'?