The cable card allows for handling of premium content that is normally unscrambled by a cable TV set top box so that you can view such stuff without also needing such a set top box. If you aren't worried about recording any such programs, you can run the cable from the wall to the VCR, and then from the VCR's coax output to the cable card equipped TV just as you are doing with your current TV. If you subscribe to premium services, your TV will be able to view those via the processing done in the cable card, but of course the VCR will see only the scrambled signal.
Most of the newer HDTV receivers have HDMI or DVI outputs (HDMI is backwards compatible with home theater style DVI equipment using a simple adapter plug or cable, except that DVI requires a separate audio connection). You will also find the newest, resolution up-converting DVD players from companies like Pioneer Elite, Denon, and Sony also have HDMI or DVI outputs. You are unlikely to find a VCR with a digital video output.
Typically you would run the DVI or HDMI cable directly from the source device to the display. If you have more than one HDMI/DVI source device you will either need multiple HDMI/DVI inputs on your display or you will need to invest in a switcher. If both ends are HDMI, then the cable will also carry digital audio. If you are using the internal audio processing and speakers on the TV that's all you need. If you have a separate surround sound processor then you ignore the audio carried in the HDMI cable and make a separate audio connection from each source device to your AV Receiver or other audio surround sound processor.