Well, the main thing is your budget. (don't be ashamed if its fairly low.) Also, is your computer close to your TV or not? If your computer is close to your TV, I would say get a capture card. That way, you can split the audio and video from your xbox to your tv and capture card, giving you excellent quality.
If the tv and computer are far away from eachother, get the DVD recorder. Record all your footage to DVDs, then play them on your computer, and copy them to your hard drive.
For capture cards, here are some good ones:
Dazzle 90
ADS Instant TV (will require an audio adapter like
this)
Dazzle 150
Note: The Dazzle 150 has everything you need, so you would not need to split the audio or video.