Hey, welcome to Cruisemates.
Check out the thread for first-time cruisers/cruising handbook, and the new posters thread in the stickies. You'll find those really helpful.
1. Meeting people onboard:
Mariner of the Seas has Fuel/The Living Room onboard (teen "clubs"). You'll find most people your age flock to those two. Fuel is the "club", while The Living Room is more of a hangout spot. You'll meet people in both. I'd suggest getting to them on the first night... cruises are shockingly cliquey. But, if you do make it, you'll meet a ton of people.
Also remember that it's equally awkward for everyone. Break the ice with something simple... "Is this your first cruise?" "Where are you from" "What do you think of the ship?" are all really simple questions that most people will not hesitate to answer... from there, strike up a conversation.
2. What to do first?
Well that all depends what time you get onboard. Don't expect much to be happening around 1pm except lunch. You might want to try the sports courts or the pool area afterwards. Or explore the ship and find your own adventure spots. You'd be surprised how convenient it is to know the layout of your ship when you're with new friends.
3. What to expect
Expect to have fun. Keep an open mind, and you'll be the happiest person around come the last day (I'd say end of the cruise, but we all know post-cruise depression always sets in by the last morning). You can expect to meet new people and new friends. Exchange contact information, and keep in touch. Your friends now will probably stay your contacts and friends later. I still talk to people from my first teen cruise.
Hope this was helpful.