Bookshelf speakers to PC??? Help with setup.

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.

If you choose not to use a subwoofer, it won't really affect sound quality except that you will have no sound quality at all at frequencies below what your main speakers can handle. How bad will this seem to you? I don't know, as it depends on what you listen to, how much you like bass, etc. etc. I'm going to guess that the quality of everything that you CAN hear will be better than you had before.

You can certainly forego a sub at first, it's not like doing so will cause any damage. According to the spec sheet, the P163s dig down to about 49 Hertz, while the Logitech system gets to about 35. So you'd be giving up a bit of your current low end extension. If you're a massive fan of dubstep then this might be a letdown. If you're a fan of flute solos it would make no difference at all. You can certainly accomplish this as you stated, with red/white connections and the Tripath amplifier.