This is from memory and it is based on the ships I spent time on. So if your ship differed, feel free to comment.
Sunrise- Topside lighting was turned off.
0600- Reveille. The Petty Officer of the Watch would turn on the 1MC (announcing system) to the common spaces and engineering and announce: "Reveille, reveille. All hands heave out and trice up. The smoking lamp is lit in all authorized spaces.." he would flip on the switch to Officers' Country and add "now reveille."
Of course, some people had been up for awhile. The watches, whether the ship was cold iron (hotel services being provided from the pier) or auxiliary steaming were manned. The cooks had been up for awhile preparing breakfast.
0615- "Early chow". The watchstanders, those who were taking the forenoon watch at 0700, got to eat. Lots of other people did, as well. Most of the officers drifted in by 0630 so they could change into working uniform and grab breakfast. The department heads and some of the division officers received copies of relevant message traffic; you could find them at the wardroom table, eating breakfast and scanning through the pile of copies of radioteletype messages. (Radio Central usually had more than one damn big xerox machine.)
0615- Sweepers. The offgoing duty section was supposed to give a quick cleaning of the ship. If the day was a weekend, it had better be a good one, as the oncoming Command Duty Officer would not release the offgoing duty section until the ship was cleaned to his or her satisfaction. The offgoing duty section hoped like hell that the oncoming CDO was badly hung over and wouldn't have noticed if the ship had sunk at the pier.
0630- Breakfast for the Crew. By now there was a steady parade of ship's company across the Quarterdeck. An officer from the off-going duty section had to eat breakfast on the Mess Deck in order to fill out a report.
0700- Relieve the Watch. The Forenoon watch was the first one for the oncoming duty section.
0700-0715- Expiration of Liberty. The exact time depended on the ship.
0720- Officers' Call. This was where the XO amplified on whatever was in the Plan of the Day. The XO would ask questions of the attendees and discuss anything that was on his or her mind. Depending on what was bugging the XO, this could be a very uncomfortable event.
On smaller ships, all of the officers and the command master chief attended Officers' Call. On larger ships, it was the CMC and the department heads; in that event, the department heads then met with their division officers. Any discomfort from the XO would be passed along.
0730- Quarters. All hands not on watch in port would assemble at their divisional mustering point. Attendance was taken and the Muster Report would later be turned into the Ship's Office. Attention would be called with words ranging from the formal (and prescribed) "Attention to Quarters", to, as I heard one Master Chief say: "Listen Up, You Varmints." The Plan of the Day was read out and then everyone would wait for the division officer to come back from Officer's Call and announce any modifications to the POD.
If the day was a weekend, the ongoing and offgoing duty sections would muster in one location. When the oncoming CDO was satisfied, liberty call would occur for the offgoing duty section.
0745- Turn To. Commence Ship's Work. In other words, get busy. Division officers were expected to be in their spaces for most of the work day. Good department heads toured their spaces as well, for on a warship, the best fertilizer was the farmer's shadow.
0800- Colors. The National Ensign and the Jack were raised. Either the colors bugle call was played (by a recording) or the National Anthem. If you were topside, you stood at attention, faced aft and saluted.If there were more than one nation's warship in port, that anthem was played as well. In a NATO task group, that could be several of them. You really didn't want to be caught topside for morning colors.
0600-0930- CO arrives on board. If the ship had no special inspections or other looming catastrophes, it was a sign of a well-run ship (and a competent XO) that the Captain arrived late and could go home early. The XO would meet with the CO soon after the CO arrived to discuss the daily message traffic and whatever else was going on. The CO would also tour parts of his ship.
1000- XO's Inspection of Messing and Berthing. The XO, the CMC and a yeoman (who took notes) would inspect the berthing compartments, the heads, the galley, the scullery and the messing areas. This was very much a "shit flows downhill" inspection, as the XO would pour hell on the department heads. Department heads would then motivate their division officers, and so forth. After a couple of bad reports, an irritated department head might order the responsible division office to make a personal inspection prior to the XO and report back.
1115- Early Chow (lunch)
1130-1230 Lunch. (Some ships called this "dinner"). The lunch hour was popular for those who wanted to work out or rack out. This hour was damn near sacred, bothering a person at lunch was considered to be an offense. Usually, immediately before lunch, the Officer of the Deck would send the Messenger of the Watch to make the Twelve O'Clock Reports to the Captain. These included the Fuel and Water Report, the Muster Report, the Mazagine Temperature Report and the Chronometer Report. The MOW had a spiel to recite as the reports were handed to the Captain.
1230- Commence Ship's work.
1600- Liberty Call. This may be delayed on ships that had a lot to do. Individual divisions could let people go earlier or later. Usually the last people off the ship were the engineering officers. 
1715- Early Chow.
1730- Supper for the Crew.
Sunset- Colors. The National Ensign and Jack were lowered, the topside lights were turned on.
1830- Muster of Extra Duty Men with the Master of Arms. These were the fuckups who had been awarded extra duty at Captain's Mast. Divisions would request extra-duty men from the Master of Arms. Usually, this involved chipping paint in either the engineering spaces or the topside weather decks for two hours.
1900 (or so, it could be later)- Movie Call. This was more popular back in the days of real 16mm movies being shown on the ships. In the 1980s, the movies began to be replaced with tapes that were played either where a TV set was installed or on the ship's internal cable television system (SITE TV).
1930- Eight O'Clock Reports- The duty department heads would meet with the CDO to report that the ship had been cleaned. Underway, the meeting was between the department heads and the XO. If a department head was on watch, a division officer would attend. Eight O'Clock Reports were really disliked if one had the midwatch.
2200- Taps. The smoking lamp was extinguished, the interior of the ship was darkened, at least in the vicinity of the berthing compartments and in Officers Country. Before the adoption of privacy curtains, all bunk lights had to be shut off.
 The converse was also true. Captains were "bonged aboard" (page down to "boat gongs"), so it was common knowledge when a captain arrived or left the ship.
 A division which didn't seem to grasp the importance of complying might end up with their division officer and chief being ordered to attend the XO's inspection. That usually worked wonders.
 "Good morning, Sir. The Officer of the Deck sends his respects and reports the approach of the hour of twelve o'clock. All chronometers have been wound and compared. Request permission to strike eight bells on time, sir." The CO would reply: "Very well, permission granted. Carry on."
 The operations officers often were heard to complain about the commuter traffic on and of the base. The engineers had no idea what their beef was, for traffic was usually light by the time they went home.
A Blast From the Past
1 hour ago