Mesh with Triangulate Modifier mesh causes new faces and vertices to form with the Subsurface Modifier and Bevel Modifier OLD TITLE: Smooth mesh to hide the low poly subdivisions

Title of the question changed, as the main issue seems to be that with the triangulate modifier and the subdivision modifier, there are new faces and vertices being created.

New question:

Using the Bevel Modifier, Triangulate Modifier, and the Subdivision Modifier, new vertices are being formed as you can see in the second image. I would like it to look like the first image, except with the Subdivision Surface modifier working, so the cutout loos smooth.

Beveling with no Limit only shows these two problems:



New Blend File:





Old question:

I created a cutout of a cylinder in my mesh. I beveled it, and it gives me this:



I would like to 'smooth' the edges to look like a nice curve, and not blocky. 

I used a Subdivision Surface modifier, but:


I already tried going around using Ctrl + T, but I could not get rid of them all, and it made my mesh look messy.

Is there a better way for me to subdivide it without the vertices going nuts?



EDIT:

R.M's method below works up to an extent. It curves the cutout, but the face deforms appear still. Is this more of an issue with my mesh?



With no Bevel Limit, the corners are sharp at the front of the mesh, but the bottom still has the corners warping in. With the Angle, it makes the mesh more round/smoother, how I want it, but the bottom still has the mesh warps. As well as the front corners.
If you don't want to triangulate the faces manually, you can use the triangulate modifier. Then, add a bevel modifier, and a subdivision surface modifier:



This gives the following result:



Be sure to triangulate before subdividing!

I'm noticing that some of the circles are still "blocky". To fix this, try to go back, before you beveled it, and use a bevel modifier (not to be confused with the one above), setting a limit on the angle. Example:

No limit:



With limit (note that the less pointy edges were not affected, and will be smoothed out by a subdivision surface modifier):