Best Display method

HDMI and DVI should provide the same picture. Both are "better" then VGA. I would get whichever digital cable is cheapest.

You should NOT use VGA. VGA is analog. On your video card all the data is processed in digital then at the output instead of going to DVI/HDMI it first goest to a D/A (Digital to Analog) chip. It reaches the TV via the VGA cable then is converted BACK to digital via an A/D chip.

HDMI vs DVI:
HDMI is DVI + AUDIO. The video portion is NO different.

The only reason to get the HDMI cable is to use the built-in speakers. In my opinion NONE of the speakers in a flat monitor are very good.

I would definitely be inclined to buy a second set of speakers. As for the DVI or HDMI cable you can still use the HDMI cable and turn the speakers off, but I assume you have a DVI cable already.

Summary:
I'd use the DVI cable and buy a nice set of speakers. They vary in quality but even a $40 set of 2.0 speakers is likely to be superior. Audio quality also depends on the sound card or onboard sound chip. An example of really good sound quality is:
Auzentech X-Fi Forte + M-Audio AV40

HDMI Cable:
I've purchased an excellent HDMI v1.3b online for $11 (plus shipping). There is NO difference in the signal as it is digital. Don't be fooled by $180 HDMI cables.
This is a Canadian site, but use the price as a reference and shop around:
http://www.cablesalescanada.com/

The problem with HDMI is that, unless you have one laying around for the price of the cable and shipping it's likely you can walk down to a local store and get a cheap set of speakers with superior sound to the ones in your monitor.