I plan to build a bookcase in a couple of weeks, I am leaning to using grade A maple plywood, and hardwood face-frame any suggestions of other plywood grades or materials I should consider without breaking the bank.

Thanks

I think your selection of plywood is a good choice. I used the same material a few years ago to build kitchen cabinets. My only suggestion is to carefully observe the thickness of each sheet if you are making dados or rabbets. Plywood thicknesses vary, sometimes even from end to end on the same sheet and plywood is ALWAYS made undersized to allow for wood movement.

Since I will be using 3 sheets,my concern also is various thickness between plywood sheets and thickness of last sheet of veneer applied to plywood I live in Southeast Florida and our resources are limited for supplies.

I would expect the thicknesses to be pretty similar, especially if you are purchasing them from the same place at the same time. Chances are they are all from the same run and should be quite close in tolerances.