IHF, I didn’t mean to overwhelm you. I’d start with your family computer by putting your photos from there on to an external without any thought to organization. It can then be attached to the new computer when you start going through them, knowing that they are still on the family computer if either one crashes. I’d also consider having a copy on the new computer as well since many photographers I’ve talked to or read about advocate having at least 2 backups of everything, even to a couple USB drives when your memory cards fill up while in the field. Then as your organizing continues, you can copy your efforts to 2 other sources. Using a backup service is a good idea although it can literally take days to upload everything the first time if you have thousands of files, but most can be stopped even in the middle of copying a file because they can remain in sync as the service determines there are new files to backup.
I also organize by year, having only 12 folders of personal photos taken from 2001 to 2012. In addition to the backups I mentioned before, having a hard copy thumbnail catalogue in addition to one created in Lightroom with keywords adds a bit of sanity to my organization efforts and makes images much easier to find.
If you have any questions, don’t hesitate to ask.