How can I make the subdivision cleaner?

subsurf gives bad results when not clean topology. like any tool you have to understand what it does. it tries to subdivide every edge to the amount you have set it up. 

subsurf works best with quad only topology, that said, you can use it with tris and ngons even, however you need to be mindful when doing so. 

best examples is that you have single plane or box and put subsurf on that and see how it divides the mesh. you can also just make a plane with 3 vertex and subdivide that and see how blender will do it. 

looking at the pic, to clean things up, you'd need to add geometry to control the subdivision.

I'm using simple subdivision, not a modifier, so it doesn't change the shape.

it doesnt change the subdivide behaviour, the edges are divided same way without altering the shape. thus you are gettin the result you have. below is cube, plane and triangle and ngon with simple subdivide set to 2.  see how the modifier is subdividing each edge and how the topology is affected.

I already did.  Eventually, I decided to just apply a subdivision surface modifier set to simple.  It did the exact same thing except without the weird gaps.