It is important to keep the hay dry. A car port probably isn't best since it does rain a lot where you are. The shed should be ok. Keep it off the floor, you can use some 2X4s to raise it up, you could use something else as well as long as it provides some good ventilation.
I keep full bales in the garage on some 2X4's and they are stacked about 3-4 high. I do store for some other people, so it's not all mine. When I need a new bale, I put it in a big rubbermaid tub without the lid and keep that in the house. A bale only lasts about a month now. For day to day use, I have another bin that holds a couple flakes and it easy to move around the house.
You can get bale bags that will fit a full or half bale. They are made out of a nylon type material, so the hay can breathe and there are some holes as well. They are easier to carry and will hold the hay even after you open the bale without too much mess. They can be a bit pricey to buy, but can be worth it especially if you can find a good price.