Copper side diagram vs component side diagram
You draw a circuit diagram (also called a schematic diagram.)  This is a drawing that shows the components and how they are connected, with a view towards describing and understanding what the circuit does and how it does it.

It doesn't care about copper side or component side.  

From the circuit diagram you make the PCB layout.  This is like a wiring and placement drawing for the real physical parts and connections.  It is pretty much a description of how to build the circuit.

In the layout, you place the parts and the copper traces to connect them.  While doing the layout you can decide which parts or traces go on the component side and which go on the copperside.

Many of the parts have a preference for component side or copper side.  Most through hole parts (parts with long wire legs) will go on the component side.  Most surface mount parts will go on the copper side.  The component library will usually have this preset, though you can change it as needed for each part.

I suggest you get KiCad to do your schematic diagram and the layout. 

Find a tutorial on the internet that explains how to make a double sided PCB.  That's an example.

The process is described in the KiCad documentation as well, but that assumes you are familiar with schematic design and layout in general.

It isn't difficult, but I don't think a "click by click" description here would be useful.  Do a couple of tutorials, then tackle your own circuit and PCB - referring to the documentation and tutorials as needed.



I've referred you to KiCad because it is free and quite capable.  There are other free programs, as well as commercial programs. 

I'm not familiar with the other free ones (except to say "stay away from Fritzing.")  Some of them may do just as well as KiCad.

I haven't used a commercial design program in many years, so I can't say much about them - except that the free and/or student versions often have artificial limitations built in.  Those limitations are there to push you into buying the full version.