A quick run down of what I would do if I were you and wanted to get a decent backup for the least amount of money.
1) With AnyDVD running, insert the DVD
2) Right click on AnyDVD and select Rip VIDEO-DVD to Harddisk...
3) Select a folder on your hard drive to rip to
4) When it's done ripping, open DVD Shrink
(I haven't used it in a LONG time so my memory may not be good here)
5) Open the DVD from the folder you ripped to in step 3
6) Reauthor as needed...remove unwanted features, languages, subs, etc.
7) Run the shrink process
8) If you have Nero installed, you can use it to burn directly from Shrink. Likewise if you have DVD Decrypter installed. Otherwise you output to a separate folder on the hard drive or an ISO image so that you can use whatever dvd burning software you wish to burn the output.
9) Enjoy your backup.
That seems like a lot of steps, especially considering you're used to a 1 click solution, BUT, the difference is you have control over the entire process. And once you do it a couple times, it becomes automatic. Of course, if you have dual layer discs like I've been getting lately, I simply open CloneCD (NOTE: CloneCD not CloneDVD...simply for reasons of Layer Break issues), read it to my external hard drive, stick a dual layer blank in my drive, and write it back out. Flawless backup every time. For shorter movies that will easily fit on a single layer disc, I use CloneDVD and remove trailers, languages, and subs to make room. If the compression comes out to less than 85% I'll skip the dual layer disc and just use a single layer backup with CloneDVD. But, you've said you don't want to purchase new software, so, for you DVD Shrink makes the most sense I think.