Michelle, this really depends on what you want to do.
If you want a finished wood with the frames on it, you can use 1/4" (hardwood of your choice here)- faced ply, or quite a few hardwoods are also available in what's called "doorskin." Doorskin is more like 1/8" in thickness, and could readily be applied without changing or removing baseboard.
If you're wanting a smooth, painted wainscot, you can use the drywall, with no backer, or again, as thin as 1/8" doorskin; Birch face will give you a tight, slick grain, oak will have more character showing through the paint.
If you don't want to remove your baseboard, I'd go with a doorskin.