For some reason, when I add the Subdvision Surface modifier, I'm getting this. Does anyone know what might be causing this?
Subdivision surface only works well with quads, and from the looks of it your model isn't constructed in the most clean way. There are a few things you can try short of remodelling the mesh:
- Remove doubles
- Recalculate Normals
- Add a bevel modifier, which might be able to clean up the edges for you so you can apply a subsurface modifier.
- Add 2 edge loops around the perimeter of the object so that subdivision works better.
My recomendation: Remodel it. Start from scratch and make sure that you are using squares to build it. Then, when the subdiv is applied it will look much cleaner and will be much easier to control. Alternatively, you can try sculpting with Dynotopo enabled (Dynamic topology). This will allow you to add exactly as much detail as you need, or even more detail than you need (which you can reduce later with a decimate modifier). Hope this helps.