You can choose from almost any hardwood or softwood.
Keep these general tips in mind:
Hardwood — is usually denser and thus more difficult to saw and frame at home. The most common hardwoods are oak, mahogany, hickory, maple, and walnut.
Softwood — is easier to cut and cheaper, which makes it ideal for DIY projects and framing. The most common softwoods (and soft hardwoods) are basswood, cedar, pine, ramin, and redwood.
Here and here are two great wood resources.
TIP: If you plan on placing your farmhouse signs outside (like on the porch or patio), make sure to pressure-treat any softwood since it absorbs moisture and can warp.