On my Harmon Kardon receiver (AVR225) I have the following choices and I would like to get opinions on rating them by superiority assuming my other components have each connection type:

Video:
S-Video
Yellow Plug (composite?)

Audio:
Optical
Coax
White/Red Plug (composite?)

My TV has an HDMI input and so does my DVD player but my receiver doesn't. Can I run video via HDMI directly to the TV and just run the audio via the receiver?

My other components are:
Digital Cable Box w/DVR
Toshiba DVD player/recorder
Panasonic 50" plasma TV (non 1080p)

This A/V novice is appreciative of any thoughts!

order of preference for connection type:

video: HDMI, VGA, component, s-video, compositite (yellow)

audio: HDMI, optical / coax, analog RCA (red/white)

That was in answer to your last question. HDMI straight to the TV is as pure as a feed can get as you're using the least amount of links in the chain. So long as you don't mind switching inputs on your TV (no biggie if your main remote has macro's).

For video, S-video is able to provide a superior picture than composite (yellow), but that depends on the TV. If your TV has a decent 3D Y/C comb filter, composite can look better.
A Y/C comb filter does much the same as an S-Video cable; splits chrominance (color) from luminance (white and darker). Not all S-video cables are equal and not all 3D Y/C comb filters are equal. Trial and error.

Most receivers don't usualy reveal a sonic difference between coaxial and optical to most ears like DACs can. Chances are you'll be equally impressed either way. In most cases the main difference is about where the digital audio cable is going. If it's being stuffed behind your gear in a cramped space, coaxial is better (optical cable can fail when bent at 90 degrees), but if it's a particularly long run, optical is likely to loose less information (if not bent at 90 degrees).
It's best to avoid generic optical and molded coaxial cables. Coaxial only needs to be 75 Ohm, well shielded with quality RCA connectors at each end. Most optical cables under $50 are a bit compromised with too much reliance on plastic optics in the tubing.

Red and white connections are simply analog audio connections (L&R). In many cases, this can be the best way to connect sound from your digital source, as many have better DACs than receivers which are priced less than $1k.

She has lovely legs from what I can see.

The only consumer device that ever had an S-Video out but stored the video in a composite format was LaserDisc. A DVD player or DVR is going to have an MPEG source and ouputing to Y/C is always going to yield a better result then decoding to composite and then running it through a comb filter.

Regardless of what specs might say there are variables.
Several years ago, when I was hooking-up satellite receivers to Sony tube TVs with 3D Y/C comb filters I began to find better results were had with a composite cable rather than an S-video cable. This was initially brought to my attention when I was hooking-up a customer's system and added an S-video for his satellite feed to replace a composite cable. When I switched the TV back on he asked me to put the composite cable back. He looked again and he asked me to leave it there. I could see why. It wasn't a poor S-video cable ($40 Monster), but when the results are better in practice I don't worry about what specs say.

I tried the same thing when I got home that day (I too had a Sony tube with a Y/C comb filter) and found the same result.

Many spec-related things in AV gear make sense in theory, but because of variables, don't always work out that way in practice.

I friend of mine who is very much into his audio gear refused to believe one CD transport could sound different to another as the info is read in binary and leaves the machine in binary and, after all, bits are bits. So I demo'd my Denon CD player and my Marantz CD player, both hooked up to the same external DAC, both with the same (rather posh) coaxial interconnect. The sound was significantly different between the two. He couldn't figure it out. He trusts and believes his ears, but wasn't comfortable with it. But he had to admit, you can't argue with results, regardless of what science tells us. To me it makes sense, because I don't believe all CD players read the information as accurately as the next. Plenty of electrical engineers think otherwise.

If your source doesn't have a lot of resolution a good comb filter can do a decent job of separating out the chrominance from the baseband signal, but it will never be as good as keeping them separate in the first place. It's analog and you can never get more from something less. You do get a slight feathering of color even with the best comb filters, so you may perceive a "smoother" image because dot crawl is pretty easy to knock out even with a mediocre comb filter.

I have no idea if there was something weird going on with the source but the only idea I would have off the top of my head is with the color feathering that happens and the macroblocking that occurs with bit starved MPEG-2 source you could of inadvertently smoothed out the macroblocking however your resolution would be compromised. A waveform monitor would of told you exactly what was going on with the signal.

These days with digital sources there is no reason not to use the best available connection available.

Particularly on some of those satellite boxes.
Something was responsible.