My older-style XBox finally died with the Red Ring Of Death (RROD). It has an external hard drive attached. The newer XBox 360s don't support that style of hard drive? I guess I could get another old one, but won't that just mean another RROD is around the corner? Are there ways of transferring the data with a transfer cable or something? Can I take the hard drive apart and somehow transmit it with my computer or something? Anything else I might be able to do?
You are correct that the older hard drives are not directly compatible with the newer models. They won't just plug in.
Microsoft sells an official Xbox 360 Transfer Cable which will allow you to attach your old hard drive to a newer Xbox 360 model and move the data over. It can be purchased online from them, or a few other places.
The full instructions to do the transfer can be found at http://support.xbox.com/en-US/xbox-360/accessories/set-up-transfer-cable