If the source is bad, it will sound bad no matter what you have on the other end for headphones. Matter of fact, it will sound worse than bad, because with a great pair of headphones/amp you will hear ALL the badness of said source. And yes, bad link in chain, bad sound.
The good news is that it is fairly cheap to get great digital sound now. Someone linked the Schiit Modi and Magni (DAC and amp) That set up gets you fantastic digital sound (relatively speaking compared to onboard). That coupled with lets say some Sennheiser 558/598's for less than 400 bucks total. I mean it sounds like a lot for sound...well it is, but once you hear it, you will be blown away. And you never have to upgrade that sound from system to system again, unlike a video card. Unless you want to of course. Great for music. Ok for gaming I should add.
The ZXr from Creative has a built in headphone amp, that can power almost any headphones. Great DAC, AND it's made for gaming.
I would snag a ZXr and some 598's (Or even better headphones). Grab everything off Amazon. With the full intention of returning it all. Play games/Listen to music for ONE day. Whatever you buy, I almost gaurantee you will keep it. Just make sure you get some good headphones of course. What do you have to lose? And if you are into it for the music, get FLAC or higher bit recordings for music.