Get the trial for AnyDVD. Use the built-in ripper within AnyDVD by right clicking on the fox icon at the bottom right, and then click Rip Video DVD to hard disc. This may be overkill, but should get rid of the copy protection. Let us know if this step fails, because then you have some trouble.
Once on the computer, you can use DVDShrink to select the sections that you want out of the dvd-video by using the ReAuthor mode and the Start/End Frames tool. Each section you cut out with DVDShrink will be a small dvd-video without loss of quality.
DVDShrink is free to download and use. http://www.videohelp.com/tools/DVD_Shrink You can find a lot of guides at that site for using Shrink.
What format do you eventually want the clips to be in?
Edit: Whappo beat me to the punch by a minute. I've used AviDemux on mpeg2 before, and don't like its tendency to lose sync. If I were going to use something other than Shrink to cut out sections, I'd use a program dedicated to cutting mpeg, like the trials for VideoReDo or one of the Womble programs.