More specifically...
A Panasonic DMR-EZ475V, a Sharp Aquos 1080p HDTV, and a Scientific Atlanta Explorer 4250HDC cable box

We have tried to use all of the diagrams provided by Panasonic to hook up the 3 devices, but none included the use of an HDMI cable. If anymore details are needed, will gladly provide.

HDMI from each to the TV. To record off cable, you can split the cable signal and have it go into the recording unit and the cable box.

If you don't get a picture on the TV from the VHS, then you may also need to run a composite audio/video cable to the TV. Composite audio/video is the yellow, white, red cables.

So in summary:

Cable box --> TV, via HDMI.
DVD/VHS --> TV, via HDMI (and possibly yellow, white, red cables)
Incoming cable signal --> 2-way Splitter --> One to the cable box, the other to the DVD/VHS.

I don't know if this will help, but I have an RF modulator that connects my TV, cable box, vcr, and video game consoles. I don't know if you can plug HD cables into this though, but they sell the rf modulators at walmart and there are several different types.

If you have an HDTV, do NOT use an RF Modulator as mentioned above. This will provide the worst possible picture and sound.

Thanks but we had everything hooked up previously, we just had to get a replacement cable box, and in doing so, some of the cables were not hooked back up correctly. We're pretty sure we had the 3 devices connected using some combination of these cables: a cable from the wall to either the DVD player or cable box, a cable connecting the cable box and DVD player, a cable into the TV from either the cable box or the DVD player (whichever is left), and then an HDMI cable likely from the cable box to the TV (input 5) and another HDMI cable from the DVD player into an HDMI switcher which is plugged into a different HDMI port on the TV (input 4).

Should just have coax from the wall to the cable box. Then HDMI from the DVD player to the TV and from the cable box to the TV. The other cables sound pointless. And there shouldn't be a need for an HDMI switcher if your TV has enough HDMI connections.