Ships move in the order that you order them but all attacks are carried out first but if you attack the same ship that attacks you in the same ordering then you both attack each other (potentially killing you both - this is the only exception to the "alternating fashion"). Base building and repair are all done after movement.
TLDR: Attacks carried out alternating, in order (except when attacking your attacker at the same time) then Movements are carried out alternating, in order then Base building, Ship and Base Repair are carried out alternating, in order.