First: whether to use familiar or formal depends on context. To your close friend, you would use the familiar and to a stranger or a person of a higher standing socially than yourself you would use the formal.
Second: To use the future Je apporterai, means I will bring a long time from now, such as next week or next month. If very near future is meant, then it would be Je vais apporter.
Thus, if talking about a plan to bring a hot drink way in the future one might say Je vous apporterai une boisson chaude; but to bring the drink in the near future, one would say Je vais vous apporter une boisson chaude.
To know why the lesson used the present tense rather than a future tense, it's necessary to know the sentence's context. Is it possible the speaker was talking at the same moment as he was bringing the drink? Thus, the bringing was happening right now and the present is correct, even though we might translate it into future in English? Finally, translating between two languages isn't an exact science because of differences in grammatical structure, and ways of expressing concepts, among other things.