There are quite a few ways you can go from here. There are hardware devices that can do this. A good example is the Casablanca video editor. They aren't that cheap however. If you want a cheaper solution, maybe video editing software on your PC will do.
When using digital camera's, you can connect the camera through a IEEE1394a (FireWire/iLink) port to your computer and digitally copy all the video / audio to the computer's harddrive. After you copied your tapes you can use video editing software to process your videos. The options with video editing software is practically unlimited. Of course it depends on the software you make use of. I really prefer Adobe's Premiere (not that cheap btw, but worth every penny). There is good (and cheaper) software though that can do quite a lot. Standard cutting, transitions with effects etc etc.... it's all possible. The same goes for the audio: you can mix multiple streams (and use transitions as well), add music, effects etc etc... the sky is the limit