Wow, what a treasure!!! I'm sorry to hear about your great-grandmother. What a tragedy. And yet how wonderful that you are willing to try and finish her quilt. It will truly become a treasured heirloom!
They look like basting stitches to me. My question is, do you want to hand quilt it or machine quilt it. If you want to machine quilt it, you can use a walking foot to stitch in the ditch (following the seam lines exactly) or you can stitch on the diagonal (making the quilt stitches from one corner to the opposite corner) or you can stitch 1/4-inch from all the seam lines. If you want to hand quilt it, there are some tutorials out there that can really help.
I'm running a bit behind and have to head off to work but I know others will post and give you great advice! Good luck with it and if you feel you cannot do it, perhaps someone nearby who quilts would be willing to finish it for you.