When Rosh Ha-shana is on Shabbat, we do not blow the shofar, as explained below (4:9-10).
Some say that if Rosh Ha-shana is on Shabbat we do not recite Avinu Malkeinu because it is inappropriate to pray on Shabbat for weekday needs, using a prayer that was originally instituted for fast days (Ran; Rema 584:1; Maharitz). In practice, Ashkenazim, Yemenites, and some Sephardim refrain from reciting it on Shabbat. Others, however, do recite it even on Shabbat (Tashbetz). Since Arizal also maintained that it should be recited even on Shabbat, this is the common custom among Sephardim. (Regarding Shabbat Shuva, see below 5:6-7.)
One must eat three meals. If it is difficult to have the third meal, whether because people are still full from the earlier meal, or because this third meal would be too close to the upcoming meal (supper of the second day), lunch can be split in two: People can wash their hands, eat a course or two, recite Birkat Ha-mazon, wait for half an hour (during which they can study Torah or take a walk), then wash again and have the second half of the meal. (See Peninei Halakha: Berakhot ch. 12 n. 2.)
When the first day of Rosh Ha-shana is on Shabbat, many of those who normally say Tashlikh on the first day delay its recitation until the second day (Rema 583:2; MB ad loc. 8; see Kaf Ha-ḥayim ad loc. 30-34).