I hate to be frank, but, you will not get good results from that horrid lame excuse for a sound card in your machine.
On-board soundcards exsist just to allow the PC to make sound with a record input so you can use your webcam (or Skype or whatever) with sound, nothing more.
I suggest you invest in an audio interface. An "audio interface" is musician jargon, though. In a nutshell, it means a soundcard that is designed for audio production.
The cheapest one I can think of off the top of my head is the M-Audio Audiophile 24/96. It can be had for around $90-$100.
Personally, I use the M-Audio Delta 1010LT, and I have been almost completely satisfied with it. (balanced 1/4" TRS inputs instead of RCA, please. My only gripe with that card. Oh well, at least it had an XLR interface thatr I could use for my mixer...)
Installing an audio interface will involve opening the PC up. It is really easy to do, but if you don't feel comfortable doing it, call your favorite PC-knowledgeable person.
90-100$, eh? I'll look into that. I'll also see what's going on with that USB mic...
Anyway, thanks again to Souliarc, Zircon, and Taucer for the additional advise. I read Zircon's production thread like Soul suggested, and I found it to be very useful and easy to read for the most part. I understood some of the definitions, but I'll most likely give it another read while I'm recording. I especially appreciated the vocal examples here and there...I just need to figure out limiting and thresholds when it comes to a cappella remix. It also seems that I'm going to use Audacity from now on...I just downloaded it. I remember trying it a while back but I found it to be very, very in-depth. It's too overwhelming. Hopefully I can get into it some way or another.
So...any additional suggestions? I recorded a short sample of my singing via Wavepad without adjusting anything. It sounds good in my ears but I'll most likely post the link to it here to get your opinions on the recording quality.
Thanks again, guys.