To my recollection, Ships are broken down into Departments and Divisions, and baby SWO's are Division Officers., responsible for leading your Division. The jobs vary , including Weapons, Operations, and Engineering. In addition to your assigned billet, you will stand watches -- deck or engineering.
Like anything else in the Navy, you start at lower levels of responsibility and through experience, performance, and sometimes just luck by being in the right place at the right time, get assigned to positions of higher responsibility, and perhaps ultimately command of the ship. There is no single path to command, but there are certainly some preferred paths or tickets to punch enroute. I will leave that to any SWO's to provide the details.