1x2 cleats on the wall screwed to the studs works well. If the shelves aren't too long, cleats on the sides as well may be all you need for support, meaning no front posts.
3/4" plywood would be sufficient thickness, 1/2" plywood might even work with a 1x2 strong back at the front, doubling as the edging. Otherwise you can get iron-on wood edging to cover the plywood edges.
With 1x2 on the ends and along the back, if you glue and screw the shelves to the supports, it will be very solid. You could pre-finish all the parts and cut and install, then do any needed touch-up.