While we haven't been on N. Gem, we've been onboard two of her sister ships, N. Jade (in 2010) and N. Jewel (a ship tour a few years ago). The teen center is fabulous! There's a juice bar inside with lots of seating for the kids to be able to hang out and chat. There are also video game consoles along one long wall of the center. I don't know if they get Internet access in there, as well.
There is no extra cost, as far as I recall, except for Internet (which is handled separately on your own accounts).