Hi Tyler, and welcoem to cruise-chat.
It really depends upon the kid. Our son is 13 and pretty much likes to do on a cruise what he does at home -- swim, play basketball, and surprisingly, hang out with us. He also doesn't seem to mind going on tours with us. Again, this is just him -- all kids are different.
Your ship will have a teen program for kids his age. Generally, the teens just hang out together. Some ships, such as Norwegian Jewel (and I presume, Norwegian Gem) have hangout rooms specifically geared toward young teens. I'll look into Spirit.