The Sony system is something typically frowned upon. You can't get the best audio through it, and it really doesn't have the proper connectivity for a fully built theater system. It's very proprietary in the setup, and generally an all around bad choice. If the Blu-ray player in it goes bad, it's not like you just go buy a new player and everything is good to go, you have to ditch the entire system.
That said, yes, HDMI can transmit all the signals. But, that home theater in a box (HTiB) system doesn't have a HDMI input and wouldn't know what to do with it anyway.
Yes, the general thing to do, if you want good sound, is to replace the very entry level system, with a A/V receiver and separate speakers. Your current system is really just... well, it's cheap. There's nothing wrong with that if that's what you can afford, but you can get quality for a fair price and still have a much more acceptable system.
Check online with Craigslist if you can, or places like
http://www.accessories4less.com for systems which may be available at a very good price as others have upgraded.
Speakers especially are key to a good system, and a quality subwoofer often costs far more than you may spend on the receiver and the speakers at some point.
A real consideration at this point is that it doens't matter if you use S/PDIF because your audio system really can't give high quality audio the performance it deserves. Playing HD audio through $10 speakers is pointless, you aren't going to notice any real difference in quality. You need quality speaker and a decent A/V receiver to even get a boost in quality from the start. If that's not in your budget to upgrade, which is fine, then just wait. Save up and upgrade at some point in the future if you want to.
