Happy to help when I can!
You've got it right about the digital output, but I'll add one thing. It's not just a digital-to-analog conversion that's needed. You also need something that can decode the audio signal (in the case of your TV, it's almost surely Dolby Digital). Virtually all receivers these days will decode Dolby Digital, but I mention it in case you have a separate DAC that might not.
As for whether that would be better than the headphone output, that depends on which is better: the DAC in the TV or the DAC in whatever you connect that digital output to. Odds are that it would be better, though, if you were to get a middle-of-the-road receiver.
I don't have any real suggestions on speakers, but there are a number of threads here (even recent ones) where that's discussed. I'd say look around a bit and you might find some great ideas. If it were me, though, I'd probably go with a ~$300 receiver (like the Pioneer VSX-1021) and get a couple of bookshelf speakers (I hear that the Infinity Primus line is a real good value and would keep your total under $500). Hmmm, I guess that I did have a suggestion.
