How do you connect cable wires to the cable box, broadband cable internet, a VCR, and a DVD player and still be able to watch a movie on VCR but also be able to record what's on TV with the VCR.

Cable in to a splitter, 1 to Modem, 1 to cable box, out of cable box to VCR, out of VCR to TV.
Depending on Inputs on TV and Outputs on other equipment allows different and perhaps better Video.

Hooking up the DVD player deends on a lot of things. What outputs are available on the player and what inputs are available on the TV, so that's a hard question to answer.

The DVD player and TV could have HDMI connections or composite video connections, component video connections, S-video connections.

Your TV could have coax only connections and that's a problem. We also need to know what kind of tuner(s) the TV has: ATSC/QCAM (spelling)/NTSC?