1) Use the freeware DVD Decrypter
(no direct link, you must look for it as it's no longer updated) to extract the whole DVD or just specific chapters you want.
2) Use DVDx (freeware from labDV)
to convert the VOB/MPG2 file to AVI and other formats.
3) Use the free and open source VirtualDub
with the video/audio choice of Direct Stream (i.e. no quality loss)
to cut out unneeded parts (even if you originally extracted just a chapter, it doesn't mean you want all of it).
But I don't like the software from 2), so I do this instead (it's more advanced but I prefer it):
1) Using DVD Decrypter to extract the movie into VOB/MP2 and AC3.
2) Using the free and open source DGMPGDec
to convert the VOB/MP2 to D2V.
3) Using the freeware VFAPI Reader Codec + Converter
(look it up as it's no longer updated) to convert the D2V to (a dummy) AVI.
4) Using the free and open source VirtualDub
to compress the AVI/AC3 to XVID/MP3.