Welcome to the forums Henley.
AnyDVD will decrypt and rip the dvds to the hard drive for you in their entirety. You will not lose anything from the original discs except the encryption. You'll get a Video_TS folder and an empty Audio_TS folder. This is normal.
Most dvds will be too large to burn straight to a single layer dvd however. You can burn the ones that are too large to double layer dvds. We recommend Verbatim brand +R DL discs for this. Use ImgBurn (a free burning program) when burning double layer discs.
If you want to use single layer discs, and still want everything on the original dvd, you'll usually need to compress the video. I suggest using DVD Rebuilder for this process as it will give the best quality results. There is a free version of DVD Rebuilder which should be fine for this purpose.
So the process goes like this:
Decrypt and rip the entire disc to the hard drive using AnyDVD.
Examine the size of the files and see if you need a single or double layer dvd.
If you can burn to a single layer without compression, or are using a double layer blank dvd, use ImgBurn to burn the Video_TS and Audio_TS folders to your disc. ImgBurn is smart enough to recognize that you are burning dvd-video, and if you have any settings wrong it will ask to correct them for you. Let it do so.
Burn at a moderate speed. That means, with 16x blank discs, and a desktop burner, burn them at 8x or 12x speed. If using a laptop or slim drive, burn at 6x.
If you want to compress the movie, use DVD Rebuilder in full backup mode.
Hope this helps.