Ok, lets start with backing up the kids' movies. I'm assuming they are commercially made dvds. Download a free program called RipIt4Me. Found here: http://www.ripit4me.org/
Read the guide at that site, it is very thorough. In wizard mode, it is simple to use and will produce a dvd movie that is the correct size to burn to a recordable dvd disk.
You can use Nero to burn your dvd-video, so that is probably the simplest path to take.
Now for the avi movie you have. Since you already have PowerProducer, you could try it. I'm not a big fan of Cyberlink's software, other than PowerDVD, but it should be fairly simple to import the avi and convert it to dvd format. Just open the software, select dvd as the output and import the avi. Making a menu may be an option as well. There should be a guide included with the software, or you can download it from Cyberlink's online site.
There are other choices for this process of course, including software from Ulead, Nero and others. One popular program around here seems to be ConvertXtoDVD.
I'm not sure how much in the way of details you are looking for, so I may not be helping much here.
P.S. If you buy your blank dvds locally, get Verbatim brand. Safest choice for good quality.