If one is planning on eating the pomegranate, which is one of the seven species, during the meal on Rosh Hashanah, should he eat that first, or does the apple in honey come first?
It is a machlokes. Most people I know start with the apple in honey, but there are some who start with the date or pomegranate if they don’t have a date.
Halichot Shlomo (1:17) writes that one should make the HaEitz upon the dates because they’re from the seven species with which Eretz Yisrael was praised and they come before pomegranates on that list (see Shulchan Aruch Orach Chaim 211:4). Ben Ish Chai (Nitzavim #4) and Nitei Gavriel 29:4 agree. Kaf HaChaim 583:13 records the practice of some people to make a HaEitz on the apple to exempt the dates. He explains that perhaps they brought out the apple first, and in such a case, one need not wait for the dates to come out to make the bracha upon them. Rav Mordechai Lebhar (Magen Avot, Orach Chaim 583:1 fn. 472) reports how the Moroccan custom is to have the apples first and recite HaEtz on them, and that all the Machzorim list the Simanim that way. He argues that it's not an issue of Kedimah, because one doesn't really want to eat both the apple and the dates right now. This, he says, resolves Rav Shlomo Zalman's issue. Rivevot Efraim 8:558:3 writes that after making HaEitz on the dates, one may eat the apples prior to the pomegranates because the rules of precedence apply only regarding the Bracha, not regarding when each food should be eaten.