All you need to do is add a boot option, for example.
tce=hda2
Where hda2 is a partition on a hard drive.
Then when you use appbrowser, the extensioins will be stored there under a directrory named "tce" and be loaded upon next boot. Also your backup of home will be saved in same directory and named "mydata.tgz"
I would only recommend other forms of persistence after you get a feel for how Tiny Core works and only if you really need such!