SWOs now have two standard 24 month division officer tours before a three year shore tour and department head school. Your time for your first tour does not start until you finish the eight-week Basic Division Officer Course (BDOC). There are other special career paths (fleet-up, WTI), but that is the standard and most common path.
Your first tour is focused on attaining your SWO qualification. This consists of qualifying Division Officer (DIVO) Afloat, Maintenance and Material Management (3M), Small Boat Officer (Boat-O), Basic and Advanced Damage Control (DC), Anti-Terrorism Tactical Watch Officer (ATTWO), SWO Engineering, CIC Watch Officer (CICWO), and Officer of the Deck (OOD) In-port and Underway. Once you attain this stack of qualifications (each with their own PQS and boarding process), you will be up for a final SWO qualification board that encompasses all aspects.
Common first tour division officer jobs include Network Security Officer (CC DIV), Strike Officer (CM DIV), Ordnance Officer/Gunnery Officer (CG DIV), CIC Officer (OI DIV), Electronic Warfare Officer (OT DIV), First Lieutenant (OD DIV), Auxiliaries Officer (EA DIV), and Electrical Officer (EE DIV).
Common second tour division officer jobs include Fire Control Officer (CF DIV), Anti-Submarine Warfare Officer (CA DIV), Damage Control Assistant (ER DIV), Navigator (NN DIV), Anti-Terrorism Officer (no division, all hands contribute to force protection), and Training Officer (no division--sometimes a few TAD personnel). It is also possible to spend a second tour at a Destroyer Squadron staff as a staff COMMO or ASWO.
See the following for detailed explanation:
http://www.public.navy.mil/surfor/ddg77/Pages/departments.aspx
Due to stacked manning on most ships (a necessity due to JO attrition in the SWO community), many first tours will also be "assistants" to second tour or LDO/CWO division officers, e.g. CE DIVO (assistant to Electronics Maintenance Officer), Main Propulsion Officer (assistant to Main Propulsion Assistant), Repair Officer (assistant to DCA). Much of the shipboard organization is standardized per instructions like the Ship's Organization and Regulations Manual (SORM), but CO's may make additions as necessary.
Common collateral duties include Communications Security Management System Custodian (CMS Custodian), Repair Locker Officer, VBSS Boarding Officer, Public Affairs Officer, Legal Officer, Administrative Officer, Wardroom Officer, and MWR Representative. It is very common for JOs to have a couple of collaterals on top of their "day job." Some of these collaterals (COMSEC, Legal, and VBSS especially) can be a very big time sink.
With a division and a few collaterals on top of qualification requirements and eight hours of bridge watch a day, it can quickly become overwhelming for a new JO. Hence the long hours and lack of sleep.
This is from the perspective of a DDG. I suspect CGs are similar, but I have no idea how things work on LPD/LCC/LHD. It is not common for first tour SWOs to end up on CVNs.