I think you could tell if they are basting stitches or quilting stitches by looking carefully at the stitches.
Can you see any knots in the thread? If there are big knots and/or long tails that are not buried in the quilt sandwich, these are probably basting stitches. If you can't find any beginnings, they are probably quilting stitches.
Also, try measuring to see how many stitches per inch there are, and post that here. In the photo the stitches look too small for basting but quite large for quilting. However, if this was the first time your grandmother was trying to hand quilt, it's quite possible the stitches would be that big.
Are all of the stitches made with a single thread?
Is the stitching all across the quilt in a pattern -- say every 6 inches apart? If so, that is typical of basting. Sometimes basting is in a grid, sometimes it radiates out from the center of the quilt.
Was the quilt in the hoop when you found it? It is not typical to baste a quilt in a hoop; hoops are typically used to hand quilt.
Look at the underside of the quilt too. Is it neat-looking? Are there any long stitches (toe catchers) there? Larger stitches would be more like to be basting. Neat and even stitches are more likely to be quilting.
I think it would be most appropriate to hand quilt this because it is pretty clear that was your grandmother's intention. If she had wanted to machine quilt, she would have pin-basted.
OH!!! I just re-read your post and see that you mentioned that the quilt is all pinned together. In that case, the quilt has been pin-basted. Most likely the stitches are your grandmother's first attempt at hand quilting. It would certainly be in keeping with her spirit to continue hand quilting it, especially since it would be your first attempt at hand quilting too. I would just keep the stitches similar in size and spacing to your grandmother's.
For hand quilting, you do typically start in the center and work out. Outlining each square would be good. If there is a border, that would need to be quilted also. How big are the squares? If the squares are large, you may need to add quilting to keep the batting from shifting -- easiest would be a big X, but you could also do a flower or other design.
Don't worry about binding until you get the quilting done. There are Youtube videos on how to bind a quilt that are very good.