I made some wood exterior shutters (raised panel type) using fir lumber. They aren't very much like what you are planning to make, but similar. Fir is pretty stable and suitable for both indoor and outdoor work. Shutters aren't the easiest project by any means, but if you are up for a bit of a challenge, they can be a lot of fun and challenge to make. You may consider them to be a significant challenge later on in the project if you decide to make them for every window in your house and commit yourself by buying all of the lumber at once though. It would be better to do them in small batches, like for one room at a time. Then you can decide at the end of the room if you want to try again for another room.
For indoor work you can use select white pine, poplar, fir, or basswood and they would all be quite stable if kiln dried to a low moisture content before working. Oak is a bit too hard to work easily, and quite heavy. If you wanted a harder wood than pine and you intend to paint them, my recommendation would be poplar. It's also one of the least expensive choices, but not when purchased at the Borgs. My preference is poplar for any inside trim or cabinet work that will be painted, because it resists denting and chipping better than the softwoods, holds screws and hardware well, yet is easy to find, works easily, and is very stable. Rockler and Woodline are likely the best sources for the needed hardware. You might also try Woodworkers Hardware.
Charley