I would use standard construction lumber for the upright parts and framing, look for nice straight 2x4s or whatever dimensions you need, the only problem you might have is; on occasion some construction grade wood can be wet, if so stay away. My choice would be clear pine for the uprights and supports. Knotty pine for the horizontal pieces would be great. If you keep the lower out of cheaper woods, maybe you could do the top out of a nicer wood. I'm not sure I would use red oak, it has a pronounced grain that really would need to be sealed well. In an absolute perfect world with no cost considerations I would do the whole thing out of white oak, it would last forever and age to great gray/silver color. Stay away from treated it'll be too wet and is really only needed for ground contact or extended periods of being wet.
Good luck! and have fun!