When a person makes the Beracha (blessing) for the blowing of the shofar (ram's horn), he should immediately take care not to have any hefsek (interruption), especially between the Beracha and the initial blowings of the shofar. If one were to speak right between the Beracha and the blowing of the shofar, they would have to go back and repeat the Beracha because of the hefsek. Especially during the first 30 kolot (blowings), a person is forbidden to speak and interrupt because he wants to wait for those first 30 kolot to fulfill his obligation. But if he did speak in between those 30 kolot, he does not go back and repeat a Beracha.
Even after the first 30 kolot are done, one should take care not to speak all the way until the end of Musaf (additional prayer) and the repetition of the Amidah (standing prayer), so as to combine all of the different blowings of the shofar together without any interruption. However, if a person did speak, as we said, he doesn't go back and make the Beracha over again. If one heard Kaddish (sanctification) in another minyan (prayer quorum), they would answer the Kaddish or the Kedushah (holiness). If one used the restroom, they would also say Asher Yatzar (Who formed man) and not wait until the end, because then they might go ahead and forget.
So, in between the sitting and standing Tekiot (shofar blasts), it is permissible to do those types of things and even to look up a halacha (Jewish law) if a person needed to. Talmud Torah (Torah study) would also be okay. But during the first 30, for sure he shouldn't, and especially between the Beracha and the first 30, so as not to create a hefsek between the Beracha and the initial mitzvah (commandment). Have a great day.