There is quite a bit of confusion here that stems from general use of the word co-ax.
Co-axial refers to a wire type with two conductors that are separated by an air gap. The centre conductor is a straight drawn copper wire and the outer conductor is a copper braid that is wrapped (co-axially) around the plastic that forms the air gap. This is then insulated normally shielded and then covered with a further insulation.
I think when you are referring to co-ax from the old days you are talking about the cables that were made to take an over the air RF TV signal from the antenna to the TV. These cables (which were made of co-ax) used either a bnc, "F" or belling-lee type connector.They carried an RF (radio frequency) composite audio/video signal and could only support 480i video and 2 channel stereo.
The step up from this connection was to separate out the audio and video signals from RF and transport them via RCA cables using a composit video and 2 channel audio 3 piece cable (red,white and yellow) This cable did not have to be co-axial construction. This still only supported 480i and 2 - channel stereo.
The step up from here was to separate out the colour (not quite technically right but enough for a lay explanation) components of the video into red green and blue. This could be transported down a normal RCA cable. These typically had red green and blue RCA connectors. This video supported analogue video up to 1080i.
Audio continued to be transported via the red and white RCA in a 2 channel analogue format or you could use a digital format that used one RCA cable (normally orange connector) that was made out of co-axial. This gave you 5.1 channel audio in a compressed format.
S-Video tried to separate out composite video into 2 channels of color and luminance. It still only gave 480i but was better than either RF or composite.
Cutting to the chase here regarding your bedroom TV.
The best cable to use would be the HDMI. This will give you digital all the way, up to 1080p video support and digital audio all in one cable.
The next best alternative.
Use a component cable (RCA red green and blue) for the video This will support up to 1080i analogue and if your TV has a digital audio input use a digital co-ax (Orange RCA) or optical (toslink) connection. This will give you digital compressed audio support up to 5.1. If your new TV does not have a digital audio input you will need to use the 2 channel stereo input (red and white RCA)
It is probably best to buy some new cables rather than rely on the ones from the "box o'junk" as corrosion on the connectors will effect your signal transfer.
Hope this has helped.
If you want to know more about the HDMI interface and HDMI cables you can download the free 21 page ebook "HDMI Demystified" at the following link.
http://www.hdmisystems.com/