First off, let me say something that you probably already learned from your research. You cannot even begin to consider being a Surface Warfare Officer if you do not have a Bachelor's Degree. Do a search for Officer Programs and you will find many threads that discuss the various programs. That said, I'll try to answer your questions.
A SWO will report to his/her first ship shortly after being Commissioned. They will be assigned as a Division Officer in one of several divisions on the ship. This means they are in charge of anywhere from 8 to 80 people that are responsible for one aspect of shipboard operations. This could be anything from First Lieutenant on a Cruiser, in charge of the division that handles anything from small boat operations, to anchoring, to actually driving the ship; to being the Electronics Warfare Officer on an FFG, in charge of the Division responsible for analyzing Radar signals that the ship picks up. As the DivO, they will be accountable for the successful functioning of that aspect of the ship's mission. They will also be responsible for the administrative functioning of that division, which means anything from reviewing and approving PMS schedules and reports, submitting performance evaluations on the people that work for him/her, planning required training events to maintain the ship's readiness ratings and requirements, reviewing/approving/submitting watchbills, and generally taking care of your people by ensuring they have everything they need to do their jobs completely and correctly. That is your primary job. This will take up 25% of your time. You will also be standing watches.
In port you will be required to qualify as an Officer of the Deck (Inport). This is the CO's personal representative on the ship's Quarterdeck. They are the first person people coming onboard see and this sets the tone for the command as a whole. To do this well, you will need to understand the ship's visitor and security policy. As an armed watchstander, you will be required to qualify on at least the 9mm pistol. You will need to be familliar with customs and courtesies of the United States Navy (and even foreign Navies depending on your location), including ceremonies, rank and recognition, necessary honores to be rendered. You will also be required to qualify as the Duty Department Head. This person is the Department Head's representative in the duty section. They must be aware of every major evolution that is happeneing in their respective department on their duty day. Duty days vary depending on the size of the command and place in the training cycle. They can be anything from 1duty day every 8 calander days to 1 day in 3.
Underway you will be standing at least one watch a day. You will be required to qualify as a Combat Information Center Watch Officer(CICWO), responsible for the functioning of the watchstanders in CIC, ensuring they are performing their duties correctly, providing training opportunities for the watch team. You will need to know the capabilities and limitations of the ship's combat systems, including sensors, communications, aircraft and weapons. You will be required to be able to manually determine the course and speed of a radar contact, the closest point of approach (CPA) that contact has with your ship (including direction, range AND time) and be able to calculate the actions required to open or close the contact to given parameters.
You will also be required to learn to "Conn" the ship. This is the actual giving of rudder and engine commands to control where the ship goes. To do this job successfully, you will need to understand the physics behind why a ship does what it does, how different engine commands or rudder commands will affect the ship. You will also be required to demonstrate this knowledge by maneuvering the ship through a "Man Overboard" drill. Trust me as someone who has BTDT, it easy to explain the maneuver, but takes many sessions of practice to be able to actually do it correctly.. and once you have it down, you have to learn it all over again the next time because each ship (even if it is the same class) handles differently.
Moving along in your quals, you work towards becoming qualified as Officer of the Deck (Underway). This person is resonsible for the safe Navigation and Operation of the ship while underway. You need to take what you learned as the Conn and CICWO, and start to actually plan ahead; know what is on the schedule for the day, know where you need to be and when, have a plan to make it happen; be able to brief the Captain on what is going on, what the problems are or could be, what you plan to do to minimize these. The best OOD's I know were always playing the "What if..." game with their watch teams.
So on a daily basis (underway) you could start out on the mid-watch from 2200-0200, get 4 hours of sleep (maybe), up at 0600 for reveille, shower, breakfast. You had better be checking message traffic and e-mails before 0645 since I can guarantee that at Officer's Call with the XO at 0715, there will be something that will catch you by surprise that you NEED to know about.. best if you already knew.
For the rest of your morning, you will probably be studying something. As a SWO, not only will you need to qualify in different watches, you will be required to learn almost EVERYTHING about your ship (and I do mean everything.. from the Sonar Dome to the droge section of TACTAS), and at least the theory behind other Navy equipment that your ship doesn't have. As a for instance, I qual'd on a Sprucan, but that didn't stop my board from asking questions about Aircraft Carriers (explain the basic concept of a Nuclear propulsion plant), Submarines (weapons and sensor systems), Naval Aircraft (okay as an IWO my board took some pity on me here and asked about the EA-6B.. one of the other people on my ship got asked about the P-3 and another about the F-14).
You will also study the basics of ship Navigation (I took a star shot line that was actually on the right chart.. not bad for a first timer.. didn't do an entire cellestial fix.. QM1 said he didn't have all night to teach me), route planning, and weather avoidance. You will study and learn about the Supply system, especially if your division ever wants to get repair parts for the gear that WILL break. You will become familliar with the Engineering plant on your ship and be able to trace out the entire drive train, from intake to screw, the fresh water system, the fuel system, etc...
Fully 60% or better of your time will be spent in studies. The Ship's Training Officer will give you a set of 20 some odd CD-ROMs that contain the curriculum you will be expected to learn over the 18-24 months you are onboard, which is the outside maximum amount of time you have to qualify. He will be on you to monitor your progress, anywhere from weekly to daily. He's not doing it because he's a creul sadistic bastard (TrainO may be a cruel sadistic bastard but there is another reason). He's doing it because the XO is on HIS *** to ensure that YOU qualify .
Lunch is at 1100 or 1130, depending on the ship. You get a break from studying.
After lunch you probably will attack any divisional paperwork that is hanging around. Those evals I mentioned, or mid-term counsellings. Any open jobs on equipment? Better check and ensure your department head is aware of them and what your recommendations/needs are. Do you have a Maintenance spot check that needs doing? When was the last time you checked message traffic to see what's coming down the pike? Better do that now. If you have 5 free minutes, finish up whatever the study topic of the morning was.
You've got the 18-22 watch tonight, so you get early chow at 1700 since you have to be up oin watch (lets say the bridge as conn) at 1730. Before going up on watch, you visit the Engineering Control to get a feel for what's going on down in the plant? Are they planning any drills/evolutions on your watch (probably)? Stop in CIC top get a picture of what's going on in the area.. what's the traffic like? What are our assigned duties? Are we on schedule to make the next event?
Up to the bridge (you had better be early, say 1720-1725, to get a solid turn-over with the person you are relieving). Take the watch on time (1730 or 1745 depending on the ship). Stand your watch as conn, paying attention to the surface picture as you see it (both with naked eyes and radar), prompt the OOD to teach you about something.. Be ready for his questions if the engineers do drills (I liked to ask my young conns what the Engineers were doing, and how did it affect the ship's handling abilities during the Engineering Casualty drills).
2130ish...Your relief might actually be on time, usually is, but you never can tell until they get there. You do ther turnover with them. On your way to your stateroom, stop one last time to check message traffic, as a headstart on tomorrow. Also stop in your divisional spaces to see if there are any crisises that have arisen. I also used this time as a chance to get a feel for what's going on since your guys/gals will also probably be oding turn over so you can just pick stuff up out of their conversations.
Some Officers will actually go to bed at this time. Some will study more. Some will unwind, watch a movie, play a game, write e-mails. It depends on if you are up to speed on the quals, paperwork, etc. I see I missed getting a workout in somewhere.. 3 times a week is the standard we are supposed to meet.
Mrs Johnny B said I should just use 1 word.. it's "Hell". It's long hours, mostly studying and learning. I noticed I didn't mention the Navy's 3-M program much, which you will need to learn and understand; Damage Control qualifications; small boat Officer duties; colatteral duties you may be assigned (such as the ship's Legal Officer or PAO if you are on a small boy, voting assistance officer, "George" duties, "Bull" duties, etc....)
There's more I probably left out, but that's a start.