This is probably a two step process...cutting, then converting. I hope you have a Windows computer, because I don't know the Mac software as well.
You can cut out small sections of a dvd using DVDShrink in ReAuthor mode. You use the Set Start/End Frame tool to select the section you want. Shrink will output in a finished dvd-video, even for a small section. Be aware that DVDShrink will work for unencrypted dvds, or those with simple CSS encryption only. If it has something more advanced for protection, you'll need a more modern decryption program, like the trial for AnyDVD.
You can download Shrink here using the direct link: http://www.videohelp.com/tools/DVD_Shrink It is free to use.
The first few pictures in this guide show how to use the Set Start/End Frame tool: http://forum.videohelp.com/threads/296328-How-to-cut-out-clips-from-DVDs-and-convert-to-AVI-MP4-using-free-software
If you need to convert to another format after cutting out the clips, you can use something like Handbrake to convert to mp4 video. Handbrake is also free: http://www.videohelp.com/tools/HandBrake
By the way, welcome to the forum Annie.