Rooting store-bought vegetables is something I just recently learned about. It does take time to grow though and I can see this with the same vegetable growing in different stages. I have only started with the celery. I googled Growing celery from a stalk after a friend told me about it. I change the water every few days and I never fill past the root at the bottom. There are many other things you could grow besides the celery. Here is a small list: green onions, lettuce, garlic, pineapple, mushrooms, ginger, tomatoes, peppers, and herbs. Some of these will grow in water like the celery but some of will need to be planted in soil.
Here are a couple of links that can help you get started: