Here are two ordinary cubes, both with a subsurf modifier, set to level 1.
The cube on the left is actually cubic and has nicely beveled edges. Here is how it’s done.
As you can see, the left cube has some additional edges (one loop shown highlighted) positioned near the corner edges of the cube. These are put in place with the loop cut tool (Control + R)
In this image, the “strength” of the subdivision (by which I think you mean how well it bevels the edges) is adjusted by moving the added edge loops away from the corner edges. The beveling becomes larger and more blocky. You can also move the edges closer for a sharper corner.
If you set this simple demonstration up on your computer, you can experiment with adding more subdivision levels to see the effect. This is probably the most flexible way of doing this kind of work with subdivision modifier, and is less work than beveling the edges by hand, which you seem to have done in your work. There is also a bevel modifier that will automatically place the edges for you, but you have less control.
You can also experiment with ‘smooth’ or ‘flat’ shading, seen in the object tools panel in the first screen shot. All these screenshots use flat shading.