MDF would work fine for painted. I wouldn't use expensive plywood and hardwood edging for painted work because it isn't structural and it won't be seen.
To get a good finish on MDF you need to seal all cuts with bondo, or joint compound (I prefer the bondo - they have a general purpose bondo filler at HD that is very easy to work with). Skim any edges and fill any screw holes, then when it dries in a couple hours sand it smooth. Prime it with a shellac or oil-based primer, and topcoat it. If you want to route profiles, or even make it look like it has raised panels in the middle of the sheet, this is the stuff to use. It machines very easily, and you just have to skim any cuts with joint compound or bondo to get a smooth finish.
MDF costs about $25/sheet for 3/4.
If you want to be more water resistant/long-lasting, you could use MDO (medium density overlay) which is basically exterior grade plywood with resin-impregnated paper covering both sides. You would need to bondo the edges so you don't see the plys through the paint, but the surface is extremely smooth, and paints flawlessly. This stuff is expensive though. About $55 for a 3/4" sheet. Nice to work with.
You can probably get MDF from Home Depot/Lowes, but MDO will have to be found at a lumber yard. (Just a warning if you have a small-vehicle - my lumberyard sells MDF in 8x5 sheets for some reason.)