First off: I just want to hook up my computer to run on my TV for the heck of it because the monitor's much bigger and would be easier to see. So of course I have a crap video card that only has DVI and VGA (....yea).
So.

If I was running a DVI to HDMI cord, there would most likely be no sound, right? From what I've read online DVI usually doesn't carry sound, but I could be completely wrong on that. And if the DVI doesn't carry sound, could I just get an audio cord to run from the computer to my TV?

If not, I'm looking for a decent video card that also has and HDMI port. Most all of them have that nowadays anyway. But something cheap (<$60 preferrably). I mean obviously it doesn't have to work wonders. Just a nice performing card that can get the job done.
(I saw this but have no clue if it's worth it..)

Any help would be much appreciated!

You could get a 1/8" to RCA audio cable, and hope that your TV will allow you to use HDMI video with RCA audio.
If not, there are pretty cheap video cards on the market that do HDMI out.