I want just to check this as I can't see any visible tracks and it's my first time making a circuit with this board.
Which pins are connected on the Arduino Proto Shield R3?
Are the following red lines joined?
If not, what's the easiest way to connect those? Wires?
Did you check the Products page for a pdf schematic of the board? That will show what is connected.
If not, then yes, wires from hole next to the header pins to whereever you need them.
Be sure to check for interference with parts under where you are putting components.