Just got off the Carnival Glory (Eastern Carribean) and thought I would share some stuff. This was only my second cruise, so I am by no means "seasoned":
1) Consider carrying a lanyard (or similiar secure device) to keep your Sail/Sign card, soda card, picture ID). You access these often on the cruise and it is a pain to dig through your wallet each time.
2) Book a massage at the spa first
thing. After a long flight/drive, plus the stress of real life - this will relax you and get you in a cruising state of mind.
3) Try things you wouldn't normally do at home. (New foods, parasailng, etc.) You're on vacation!
4) Talk to your tour guides on excursions about their experiences. As Americans we tend to have a very narrow and limited view of the world. You'll be surprised how much they will appreciate it.
5) There will be lots of opportunities for shopping on shore. However, if you see something on the ship you want, you may want to wait until the end of the cruise. Sometimes they have sales at the end because it's their last chance to squeeze your pockets. Just make sure they have enough of whatever you're buying so you don't miss out.
6) Leave your worries at home but please don't leave your manners.
These are just a few things off the top of my head. Take them for what you will. I'm sure more experienced cruisers will have some different ideas.