It would appear you need 2 x 1/4 inch (3.5mm) jack to RCA cables, one for each channel and each speaker, left and right, rather than just the 1 cable ("..and a mini to RCA cord...") you seem to have bought.
The Yellow and Black RCA ports appear to be the Left and Right speaker channels respectively.
The output cable with a green jack that connects to the green jack on the computer's sound card seems to be tied in with the wired Remote Control.
I would have recommended getting a pair of passive 40W RMS Logitech Stereo Speakers.
As long as the speakers you have match in terms of Wattage and Impedance, you should be okay.
I hope this helps.