Is there a specific minhag regarding when and in what order to eat the simanim? In other words, is it acceptable to serve simanim as a matter of course (leeks with the steak, date cookies for dessert, etc.) or should they be served in their own special time?
The blessings are prioritised (in general, not just on RH). First "Pri ha-etz". Then "Pri ha-adamah". Then "Shehakol". In "Pri ha-etz" priority is given to "7 species" fruits. Within these, the priority is given to the closest to the word "eretz" - land. Thus the priority is olive, date, grape, fig, pomegranate. "From an article by Rav Eliezer Melamed (in Hebrew)"