Attaching a circuit on to a PCB
First, draw out a schematic for your circuit based on how you already have it connected. This shows what is connected, but it is not important how the components are placed on the breadboard; distances don't matter on a schematic. A good schematic will make it easier to understand and explain what your circuit is doing, and it will help you make the right connections on your board.

It looks like a Zero PCB is a board with holes for you to put components through and solder them. They will not be connected to any other components unless you make the connection yourself. You can place them really however you like, but certain placements will make for more convenient connections later.

You need to solder your own wires between the different holes on the bottom side of the board, or use any extra length from the leads already on the components. To solder another wire to connect two components, you can choose if you want to use bare wire (remove the insulation entirely) or keep the insulation on except for the ends. For very short connections, it's easier to just use a length of bare wire, but it's better to have insulation on longer connections so you don't have any accidental shorts to other components.

Lay out the wire between the two components you want to connect, cut it to the right length, and solder the two ends on top of the pads where the components are soldered. (You said you already know how to solder, so I won't go into detail there unless you would like more.)