Ro, Congratulations on your first cruise. I'm sure you will become addicted like the rest of us. My boyfriend and I did the 4 night cruise out of Tampa last August. We were on the Sensation and went to Cozumel and Grand Cayman. I'll try and answer your questions from what I have learned on my past cruises:
1. Before you get on the ship, you will get a Sail/Sign card. They will take your picture and keep in the computer system. The pictures is not on your card. That way if you lose your card, somebody can't pick it up and try to get on the ship with it. Also, they can keep track of who is on and off the ship. You will need to get on and off the ship in port. The card is also a credit card. You can secure this by using a credit card or using cash. I have always used a credit card and had no problem. You will use your Sail/Sign card for all purchases on the ship (i.e. gift shop, soda, alcohol, pictures, excursions, etc.) You will want to take cash with you into the ports. I used my debit card once and it was a pain. Also, in Cozumel, they don't have any ATM machines by the ships. If you don't have cash, you will have to get it from the ship's ATM or go into town.
2. There will be one formal night. Usually the 2nd night of the cruise. Men can wear suits or tuxes. For women, I saw everything from really dressy dresses to just regular dresses. They don't permit shorts in the dinning room at night. Except the very first night, they are a little bit more laid back about things because some people may not get their luggage until later that night. So, whatever you wore unto the ship, you can wear that into the dinning room. After that night, it is pants for men and pants, dresses or skirts for women.
3. Bars on the ship you will use your Sail/Sign card. Some of bars in Cozumel take cash only. So be prepared to use cash in port.
4. Cozumel was awesome. My boyfriend and I went to Carlos -n- Charlies and had a great time. We also did some shopping and then stayed at the bars by the ship. You can take a taxi into town for little or nothing. They have great deals on a lot of things. Just be prepared for the local merchants to hassle you to come and look at their shop. If you are not interested, then just say "No thank you" and move one.
5. I never had any problems with valuables and I never used the safe. I just kept all my stuff hidden in luggage and nobody ever bothered them. I know the rooms do have safes for people to use.
Hope that helps. Have a great time on your cruise.