If you go here
there are two itineraries of the cruise compass from October 2009. Not sure what activities / shows that you are interested in.
The evening shows (production showtimes) are free. What time you go depends on what dinner seating you have, with the exception of the first night. That night there is usually one show for all guests. Don't worry. If you have the late seating for dinner, the show will be done in time. On my last cruise on the Enchantment the cruise director stated "We learned this a long time ago. You never ever, interfere with food."
The only thing you may need to get tickets for, but I believe there is no cost for, is the ice skating show. Look in your cruise compass that for where they will be giving out tickets. It's usually on the first full day of the cruise, when you are at sea.
There is a charge for the Bingo sessions, which is somewhere around $35 per session I think. However, you cannot use cash for this.
Royal Caribbean uses a cashless system. Your room key also acts as your onboard spending account. Anytime you make a purchase they will ask for your SeaPass card. Whether it's wine at dinner, shopping on the Promenade, etc.