DSynchronize is what I ended up using after trying out and comparing quite a few different backuping software. I've been using it for several years now.
It's a small program with a minimalistic design.
Still it can do pretty much anything one could wish for.
You basically pair sets of folders. This can also be done to a remote server via FTP.
There are some settings that can be changed, for example regarding whether you want them synced both ways, or whether to run a more standard back-up by copying files one-way. And plenty of more.
You can schedule automatic backups. And also add filtering conditions.
When files are deleted or replaced they can be moved to a "trash-can" folder for a while before being removed indefinitely.
But the thing that makes it outstanding in my opinion is the capacity for "Real-Time sync". DSynchronize senses within seconds whenever a file has been added/changed/deleted in any of all the folders it overlooks and then imminently performs whatever sync/back-up action it has been instructed.
Setting up a system for back-up/syncing that immediately updates a an exactly mirrored version of selected parts of your harddrive on an remote server in a different location must be the most foolproof way for keeping all your digital stuff safe.