Your Logitech setup had an amplifier built into it, it was not just a set of speakers. You need to duplicate this concept when moving to the passive speakers you are considering. You don't "need" a new sound card, although adding one to your system can give you more options.
The tripath amplifier, an AV Receiver, or a stereo receiver with bass management (rarer, it's probably easier and cheaper to just head straight to the multichannel AVRs) will be a required component to take the low level signal output by your computer and turn it into something your speakers can actually play.
As for how things hook up, that will depend on so much. If your computer supports HDMI output your best path to full fidelity is probably going to be HDMI output from computer (configured of course, to send audio that way) to a receiver, HDMI from your receiver to your monitor, and your two main speakers attached to the AVRs left/right speaker terminals. The RCA subwoofer (.1 or LFE) output on your AVR will plug (via a 'digital audio' or 'composite video' cable) into your subwoofer's input, or often the left input if there are two.
Without HDMI, you will essentially do what you did with the Logitech system, except that you'd probably need to add a splitter and adapter to turn the 3.5 mm stereo into twin RCA left/right. Optical or coaxial SPDIF would be better, because it supports multiple channels. HDMI would be best, because it supports transmission of lossless multichannel audio.