Without using any power tools these are the tools you could use to do this:
- scratch stock
- rasp and/or file
- card scraper
- chisel or gouge
Note that sometimes a combination would be used, rather than the entire job being done using the one tool. Also note these are in rough order of preference.
I haven't listed sandpaper here but the entire operation can be done quite well using just sandpaper, particularly if a modest rounding (small radius) is desired. It is also routine to use some sanding to refine and smooth a rounded edge produced using hand tools.
A moulding plane with a suitable bit would be hands-down the best tool for this job but as they're a specialist piece of kit I'll discount them. So, in general the simplest and easiest handtool method to round an edge is to first create a 45° chamfer (often done with a block plane but in theory nearly any hand plane can be used) and then to knock the corners off the chamfer with one or two light passes of the plane, in essence creating a triple chamfer.
[Source: Choosing & Using Hand Tools by Andy Rae]
This 'faceted' edge can then easily be made smoothly round with minimal scraping or sanding.
Note that for this project because of the grain orientation you're probably better off if you confine yourself to doing this with sandpaper. Start with 80 or 100 grit, then 150, then finish off using 220 (or whatever the final grit is that you used on the face of the board).
Traditionally this would be done with a chisel, although in theory hand planing using the above method could also be used if the workpiece can be secured at a suitable angle.
The usual chiselling technique would be to have the workpiece flat on the bench and to pare downward with the chisel:
Note the grip used and the position of the head so the user can sight down the length of the chisel, this is the textbook stance and hold for vertical paring and it allows very precise, controlled shaving to be done.
Finger grip cutouts
There are a few different methods that can be used here. I think your simplest option would be to drill two holes* for the rounded corners followed by sawing the 'shoulders' and then chopping back to your marked line with a chisel. Final flattening/smoothing of the bottom of the groove (what will be the top with the board flipped over in use) would be done by careful horizontal paring using the chisel, or with a hand router of one kind or another.
*If only hand tools were available here as well, this would usually mean using a swing brace and suitable bit; a wheel brace is not usually powerful enough to drill holes of a greater diameter than approximately 6mm / 1/4".