For reference, I can (finally) get a good instrument recording onto my computer.
My new amp has a DI (Direct In). You can also get stand alone DI boxes (SansAmp BDDI for example).
So, from my amp, the DI goes to a 25' XLR microphone cable, which goes to a XLR-1/8" Stereo plug. This is plugged into my soundcard.
Granted, this is not a pro setup by anymeans, but it works well, and quite clearly. The biggest thing is that it bypasses the very noisy preamps on the soundcard. I can adjust the DI level from the amp (or DI box) so I can get the level. Previously, going from 1/4" instrument cable into the soundcard didn't work well, way too much noise, due to preamps.
So a basic recording-bass-on-your-computer setup would be a soundcard (obviously), an XLR-1/8" connector ($10), XLR microphone cable ($20), a form of DI box ($?)