You can do either. In the pot in soil helps protect the tuber from temperature change. But it is good to give it fresh rich soil for the new season. They can be dug out and stored in a dry cool (warm if you're in a very cold climate) place. In the new season they begin to sprout and that's the time to plant them again. Some people leave them out and wait for the flower as it looks unusual to see a flower on a large tuber sitting on a shelf. But that depletes the tuber which only has stored reserves to grow on rather than gaining from the soil.
I'm in Central FL. So I think I could leave it in the pot then, and store it in the barn, where it stays dry and rarely goes below 40º.
Do I plant the little bulblets now and store them the same way ?
You don't have to plant the bulblets, but remember they dry out much faster than the larger tubers. I keep them in a plastic bag with a damp (not wet) piece of coconut husk. Anything that'll stop humidity levels from getting too low. And they're more susceptible to cold because of their smaller size.