The hazara (repetition) of Musaf on Rosh Hashana is very significant. The Hachamim established that teki’ot should be blown during each of the three middle berachot: malchuyot, zichronot and shofarot. According to some Poskim, these are the main teki’ot of the day; the teki’ot blown during the silent Amida are only a custom. Therefore, it is crucial to listen to the hazara, and to the teki’ot, as the teki’ot are an integral part of the berachot.
Some have the custom to stand during the entire hazara. They bring a proof from the Rambam (Hilkhot Tefilla 9:3), who writes that "hakol omdim" – all stand, listen and answer amen. Others, such as the Yaskil Avdi, understood the Rambam as meaning that people were silent , but not necessarily that they would stand.
It is permitted to say the hazara silently, with the Hazan, in order to maintain focus and kavana, as long as he does not disturb others. However, he should not say the name of God during the blessings.
After the Kaddish Titkabel, another ten teki’ot are blown, and then after Musaf, a teru’a gedola is blown. Some say to blow the teru’a gedola after Aleinu, others write that the teru’a gedola is blown after Barechu, some say before Barechu. Our custom is to blow the shofar after barechu, before Aleinu Leshabeah. The reason for this extra sound is to confuse the Satan, so that he will not accuse the Jewish people of poor behavior as they leave the synagogue and go home to eat their meal.
Summary: One should listen attentively to the berachot, and the teki’ot, for the entire hazara. After Musaf, in addition to the last ten sounds, a tefi’a gedola is sounded, before Aleinu.