These days, boarding a ship is similar to boarding an airplane.
Typically you arrive at the terminal, and drop off the luggage which you don't plan on carrying on board with you. From there, you go through x-ray screening and passport control. Once through, there, you go to a ticketing agent, where you are checked-in and given your room key. Once you are checked-in, it's off to the gangway. You present your room key at the gangway entrance (when boarding most ships, you are entering foreign territory.) At some point aloong the gangway, you will be stopped for pictures, (expect to see a lot of photographers throughout your cruise), and perhaps given the chance to sign up for some kind of soft-drink or wine plan.
Typically, from the gangway, you will board the ship at the ship central Atrium or lobby. There will be numerous ship's personnel on hand to great you and offer directions or other assistance.
Depending on how early you arrive, your cabin may or may not be ready yet. If the cabin is ready, you can drop off any carry-on luggage, and head to the buffet for lunch, head to the excusion desk, change into your swimsuit, or start exploring the ship. If the room is not ready, you will have to carry it around with you for a little while. At some point, while wandering around deck, you'll be offered a drink, typically in a "collectors" cup. Just be warned, when asked "Would you like a drink?" they are really asking "Would you like to buy a drink?" Most of the food on cruise ships is free. Soft drinks and adult beverages are not.
Cruise ships will often let folks start boarding around noon, with the final boarding call around 5:00pm. Around 5:30pm or so, there is a mandatory lifeboat drill. The life boat drill requires you to don your lifejack, and head down to the designated deck, and hang around for a little while. Once that's done, you can head back to your room, drop off your life jacket, and head up to the upper decks for the sail-away party. Or, if you have a room with a balcony, you can hang out there.
The casinos typically don't open until 7:00pm or 8:00pm.
Hope this helps.