Hi baby_g, welcome to the DR1 board!
As people asked, it would depend alot on where you are moving. DR is a large island and living in say Santo Domingo, or countryside or say in Sosua or Puerto Plata are all very, very different.
And teens would do different things.
Where we live, in Puerto Plata, it also depends on if you are a girl or boy. My guess by your username, would be that you´re a girl.
Many kids have afternoon school as well ( in ENglish ), the boys play baseball, basketball, they go to Game clubs or internet centers. The listen to alots of music, rent dvds etc, watch baseball and of course study alot at that age.
My son, who is 16, says the following about the girls and what they do:
They hang, walk around, gossip, do windowshopping, do nails both at places or help each other. Visit each other. Have afternoon school.
There is also a huge difference where you live, i.e in the city, outside the city etc. If you speak Spanish or not. If your friends would be from a gringo community or the local teens.
Tell us more as of where you will live, what school you will attend etc, and you will for sure get more replies.
Edited to include:
I guess it also depends on why you are moving here. If you are moving because a parent got a job assignment, you will most probably get help and support as well as advice from the company. If you´re moving to live the adventure, without really knowing alot or having been here alot, it is going to be just that - an adventure.
If you´re moving with a parent, because he/she found the (young ) love of her/his life at a vacation on an all-inclusive resort, it will be harder, as that is not a situation many local, decent families will want to have anything to do with.