Good choice on the trim style :)
A typical 1x4 is 3/4" x 3-1/2", and a 1x3 is 3/4" x 2-1/2".
In addition to various widths, you should have several thicknesses. The casing should be thicker than the base, and the stool will look better at least 1" thick.
Try to find a millwork supplier, or hardwood lumber supplier for your material. They can supply long lengths -- you want as few splices as possible, and can provide material any size.
You will want S4S (surfaced 4 sides) and ideally, sanded on one face. The sanding is important, especially if you are using a glossy finish. Otherwise, you may see knife marks in the wood from the sizing operation.
Consider popular as an alternative to pine.
I would ease the corners -- paint does not bond well to sharp corners, prime and sand before installation.
Q