Cutting shapes out of meshes

There must be a way to cut out shapes from an object. I have looked around and can’t seem to find the answer. What I want to do is something like this:



I want to be able to cut a circle out of a mesh, like I did in the picture above, but in a simplified and better looking way.
Here is how I did the box above,


I added my two modifiers(Bevel and Sub Surf) and deleted a vertex from the middle of the square and selected the surrounding edges and subdivided them, creating the left image. Then I re-shaped the verticies into a circle, creating the right image.
Is there any faster way to do this? And the box I made had a vertex in the center perfectly, but on this other mesh that I’m trying to make it doesn’t have a perfect vertex in the middle, so I can’t cut a hole in it right. This is what the other mesh looks like: (the part that I am trying to cut a hole into)


If I do what I did above it won’t turn out right. All help appreciated.

Either Boolean or just project a circle on the face, delete vertices then rejoin them up with the circle with edge loops to give a clean edge if you want to subsurf

Alright that fixed everything. Thank you so much! I had read something about the boolean modifier but I never thought to search up videos for it. Well I finished what I was working on so thank you!