Welcome to the forums MProcun.
Lets get a few terms defined before we go any further. When we say rip, we mean transferring the data from an optical disc to the hard drive. If you are starting with commercially made dvds, then you need a ripping program that will also decrypt the movie before it can begin to rip to the hard drive. DVDFab can be used for this purpose, and it is actually the only purpose for which I would use it. DVDShrink has the ability to decrypt dvds that only have the simple CSS encryption. Anything newer will defeat it as a ripping program.
So, the first task is to put the entire dvd on the hard drive and work with it from there. This is a fundamental step that eliminates many problems. If the clips you want are from the main movie, and you don't want any from previews, commentary or extras, you can rip just the main movie with DVDFab. And once you are done with all your work, you can delete the movie, so its not like its going to be taking up permanent space.
Ok, the next step is cutting out clips. Lets assume you want some sections from the main movie. There are three programs I use, or have used for this purpose. Sony Vegas is way overkill for this by the way. DVDShrink, as explained by cholla, is able to cut sections out for you in Reauthor mode, using the Start/End Frames tool. Each section, when saved will be a small dvd-video, which has advantages and disadvantages, depending on what you want to do with the clips.
A second tool for this is AviDemux. I rarely recommend using this tool for the mpeg2 found in dvd-video, since it has in the past often been guilty of losing sync between the audio and video. But the new version, 2.6 is better, and if you cut on I frames, it should be fine.
Neither Shrink nor AviDemux are frame-accurate cutters. If you need more precision, I would advise getting the fully functional trial of VideoReDo TV Suite. This program can cut anywhere in the time line, and will only re-encode around the cuts if necessary (smart rendering). This program is very easy to use, and can be used in Scene mode, where you set your cut marks and it saves the area between them. VideoRedo TV Suite will save as mpeg2 files, but it can also author these files to dvd-video for you.