For best results you can either combine curve circles in 2D mode, then keep it as a curve object and extrude/bevel as needed OR you can create a mesh plate circle and a group of mesh cylinders and use the boolean modifier. You won't get subsurf on the inner holes this way but it's better if the plate has to be a mesh.
Option 1 - Joined Curves in 2D Mode
Create the small holes as bezier curves as well as the large plate circle. Make sure all are in 2D mode, and then select all, including the large cirlce and press CTRL J to combine them all into one Curve object. Because you are in 2D mode the inner circles will cut out the the outer plate circle. Then you can extrude the curve and add a bevel. Keeping it in Curve mode will give you the best smoothness on the inside of the hole cutouts. If you convert back to mesh, you won't have the same level of smoothness, and adding subsurf will introduce shading errors.
Option 2 - Cylinder Group with Boolean Modifier
Create a Mesh Circle for the plate. You can extrude/bevel/subsurf it to get it looking right.
Then take your group of bezier circles, join them together into one object (we'll call it "Tubes").
Reduce the curve resolution (Preview and Render) to the lowest possible while still maintaining a circular shape (2 or 3 usually works, we need lowest geometry possible and will regain our smooth curve with subsurf later on)
Make sure the Tubes curve object is in 3D mode (this is to avoid creating caps in the mesh cylinders) and then convert the Tubes object to mesh with ALTC
In Edit Mode extrude the circles vertically so that they will be greater than the thickness of your plate mesh.
Add a Boolean modifier to the Plate mesh and select the Tubes object as the target object. Select Difference as the mode. Then set viewport and render visiblitiy of the Tubes object to off in the Outline. You'll see perfectly cut holes in your plate object. (Note: you control the smoothness of the holes with subsurf on the Tubes object)
In order to hide some viewport surface errors, select the Plate object, go to Object Data toolset, in the Normals panel, check the Autosmooth options, and change the Angle to 30 degrees. You should now have perfectly smooth surface with smooth holes cut through.