Sophia here from Shopify. Welcome to the Shopify Community, and congratulations on starting your new store!
For the product's inventory settings, do you see either Track quantity or Continue selling when out of stock selected? Sometimes a product will be listed as "SOLD OUT" without an Add to Cart button available when the inventory tracking hasn't been set up yet.
It would also be helpful to know the name of the theme you're using. If you can share your store link here, I'd be happy to take a closer look with you.