The first thing I do when I get onboard is head to the buffet for my first lunch onboard. The earlier you board the less crowded the buffet will be. It is nice being able to be finished with my food before the other cruisers have even gotten through the buffet line. Staterooms (cabins) are typically not accessible until a certain time because the stewards are still preparing the room and cleaning up after the cruise ship's last guests. Sometimes cabins will be available early if you paid for a certain program or have a perk with the cruise line. Between lunch and being able to access my cabin I usually explore the ship and walk around on the upper decks.
Before the ship sets sail there is a required safety drill for all guests. Each cabin is assigned a muster station that you will have to report to in the event of an emergency. Go to your muster station when the direction is given from the PA system. Bringing your lifejacket depends on the cruise line and ship. Usually there is no photography, videoing, food, drink, or cell phone usage allowed during the drill. I have heard of passengers getting ticketed by the Coast Guard for doing those things during the drill. I don't think that is the way anyone would want to start a cruise. Since the drill is required don't even think about hiding. Stateroom stewards check every cruise cabin, every balcony, and under the beds in the room. Some stateroom stewards stick something in the key slot on the cabin door to know the room has been checked. Not attending the drill could result in you and your cabin mates getting kicked off the ship without refund. Or another consequence that could occur is that you hold the drill up for everyone else. Whenever someone holds up the drill they become very unpopular onboard. Overall, the safety drill is for the benefit of all passengers so you know they usually become very unpopular onboard. The safety is for the benefit of all of the passengers so you will know what to do in the event of an emergency. Take it seriously even though it may seem like a waste of time. The muster station drill is my least favorite part of the cruise vacation.
After the drill you will most likely be setting sail! You are on vacation and what you do next depends on you. Most passengers usually attend a sail-a-way party (depending on the ship and line) and many also line the outside decks to watch the scenery go by as the ship departs from port.
Please keep in mind that everybody's first day of the cruise will vary. This may not reflect all cruise lines & ships. However, there are some characteristics that most cruise's first day will share.