You just are not going to get a great projection surface with plywood, imo.
However, if you are so inclined - use a medium gloss white for the wood. It's going to get a little dirty over time as the wood soaks up dirt, grease, natural oils, not to mention Coke stains, pizza, etc..
For principal use, get a presentation pad of paper, with a grid of 1" squares. They are 3x4' sheets. Should cost you $10 for a pad of 50 or so at Business Depot. Use this as your main surface on your tabletop and your painted plywood as your spillover for the projected image.
Generally speaking - keep the grid on the projection surface. You can use photoshop and some other programs to do a grid on a separate mask - but it's a pain and it's fiddly. You end up screwing around with it instead of keeping your game moving along. A pad of white paper pre-gridded with 1" square is no muss, no fuss takes exactly ZERO seconds to maintain as you project on to it - and may be marked up physically if you need to do so.
If it gets dirty over a few sessions (and over time, it will) - rip it off and a brand new gleaming white page is right there, ready for you to use. Unless you are marking up your pages multiple times during a session, a pad should last you - what - a year or three - depending on how often you game? For $10?? That's a bargain.
Note - most ceiling brackets for projectors won't work for this purpose as you need to point it straight down and they are not made to accommodate that angle; however, a WALL mount bracket for a projector works extremely well for Tabletop projection when it is mounted to the ceiling (it will permit straight down with as much of an angle as may be necessary). You can find em on eBay. (search "projector mount")