If the dvds are not encrypted, you can simply copy and paste the Video_TS folder over to the hard drive. If they are encrypted, you'll need a decryption/ripping program like DVDFab HD Decrypter.
To take sections out, you could use DVDShrink in ReAuthor mode using the Start/End Frame tool, but you want to do more than cut sections. In order to make transitions, add music and reauthor, you'll need a full fledged edition program that can handle mpeg2 video found in dvds without causing video/audio sync problems.
You should look at a complete, commercial program for this like Womble Mpeg Video Wizard DVD, Adobe Premiere Elements, VideoRedo TV Suite, or Corel VideoStudio X2. I believe all of these have free downloads to test, and see if they meet your needs.
I'm assuming you want a dvd as final product.