I have been interested in this problem for quite some time. I do not consider myself an expert, but I have worked out some methods with which I am satisfied.
1) For LPs and audio cassettes, I output the sound to a portable laptop computer that I bring close to my stereo. I have a cable from Radio Shack that converts the red and white RCA plugs to a 3mm plug, which then can plug into the audio input jack on the computer. Presently I use an Apple Powerbook. I used to use a Dell Laptop.
To record the sound, I use a wonderful digital audio software program called Total Recorder, which is made by a Canadian firm called High Criteria. It can be downloaded for about $39.
The software may have to be configured to the particular kind of audio input. The instructions can be a little bit difficult; just try it.
I press the red record button when I am ready to capture sound. It is possible with the software to observe the loudness waveforms of the sound. You can adjust the recording level with the software, or save your recording and afterward re-adjust the sound levels.
I usually let an entire LP side or tape get recorded, then I save it as an MP3 file. (It can also be saved as a PCM file, which can then be copied onto a CD if you want to play your music as a CD, but note that this takes up a lot more file space on your computer if you temporarily save it there.) After I have saved the file, I play it back and break it up into the individual songs, which I label. The visual display allows you to fast-forward through the file to see where the sound level goes down, indicating the end of the track.
With the MP3 files I have saved, I then import them to iTunes on my computer. Then they are well preserved, ready to play back any time without further wear and tear. They import well into my iPod also. I download the album art either from iTunes or from Amazon.com.

2) For video tape, I use a stand-alone DVD recorder from Toshiba. The device costs about $150. You can run the analog output from the VCR into the DVD recorder. The manual for the latest model is confusing, takes some getting used to, but it can be done. I prefer Toshiba because I can also copy my analog 8mm video tapes and MiniDV tapes from camcorders to DVD by using the Digital Video (also called Firewire) port on the front of the unit. (This is not common with other manufacturers.)
Either way, I get crisp, clean looking DVD video that I quickly copy on a coputer to keep a safe back-up copy. (This also allows me to check to be sure there has not been a flaw in the recording, which sometimes can occur.) Software can also allow you to capture digital stills from the DVD recordings. This is a great way to make family pictures that you can send around.

3) Movie film I cannot do. My step-mom sent our family 8mm movies to a service that made a VHS video copy, many years ago. Since then, I made DVD back-ups with the above method.

Note I do not use a computer for video. I have found too many problems with freezing and dropped frames, at least with PCs. I intend to learn movie editing with the Apple Powerbook, and maybe one day I will become converted to computer video recording and editing. For now though it is simple to output video from a VCR or camcorder to a tabletop DVD recorder. Do it soon, because tapes do degrade over the years.