I'd also check out the Matrix M-Stage. It's been getting fantastic reviews, and I just picked one up (lightly used) for $210 shipped.
NOTE: All of the following is presented as my own summary of many, many threads that I have read here on Head-Fi over the past month or so. Being a fellow newbie, albeit already obsessed, I just wanted to share whatever I've picked up so far as I know it's sort of tough to get started on all this new info at the very beginning.
I'm not really an expert on your particular soundcard, but it's highly unlikely from what I've heard that it's putting out a clean analog signal at 300 ohms, which is what the impedance of the HD650s are. Realistically, to take your listening experience "to the next level" (as cliché as that sounds...), what you'll want to improve is (1) your amplifier, and hook up your sound card's line out to that. (2) Next, you'll want to move your DAC (Digital to Analog Converter, which essentially the function of your soundcard + its crappy built in amp) outside of your computer, where it will be free of any noise, etc. produced by all the circuitry inside your computer case. And finally, once your budget allows it, (3) improve the cabling between your components. Of course, the order of 1, 2, and 3, from what I understand, is absolutely arbitrary, and there are lots of steps in between an M-Stage and a truly top of the line amplifier (Balanced Beta22's seem to be regarded as the top of the line SS amps, and a range of tube amps seem to be in contention for "best tube amp ever").
So to sum up, I'd highly recommend getting an amp next. The big choice you'll have to make is whether or not to go tube or solid state. I'd recommend starting off with solid state, just because many agree that the quality of "entry level" solid state amps, especially contenders like the M-Stage or the Asgard, is remarkably above their price point and just plain very good. As an added bonus for you, both of these amps are especially good with the HD650s. Schiit claims on their FAQ that the Asgard sounds especially good when paired with Senns (and the HD650s in particular), and the M-Stage is basically a part for part clone of the Lehmann Black Cube Linear, which was designed with Senn HD6X0 cans in mind. Tube amps, on the other hand, require a considerably larger investment, and many say that the intro ones (such as the DV 336SE) are good, but need to be "tube rolled" (the tubes need to be replaced) in order for them to sound great.
I'm JUST starting into this hobby now too after sort of wanting to dabble for years (just ordered a pair of HD650s and an M-Stage to go with my uDac-2), and actually placed an order for a DV 336SE on ebay like 4 days ago, but cancelled the order before it shipped because I've been doing all this obsessive reading lol.
Anyways... that's the gist of all the info I've gotten so far. Happy listening, dude, and report back whatever you hear. At the end of the day, your ears are your best friends and the most important listeners, so trust them over the online ramblings of headphone obsessed crackpots
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