Are the dvds you have now actually dvd-video? (Video_TS folder containing .ifo, .bup and .vob files).
If so you don’t necessarily need to convert. There are a few editing programs that can handle dvd-video fairly well, and though the better ones are not free, they do have fully functional trials for you to test them for a couple of weeks. Look at VideoReDo TV Suite, or Womble Mpeg Video Wizard DVD.
Edit: And to answer your actual question, if you still want to convert to avi, you can use any one of a number of different programs. AviDemux, FairUse Wizard, AutoGK…all can convert dvd video to avi.