How can I see unused gift card balances across my store?

Go to Analytics → Reports and open Outstanding gift card balance (in the Finances category, or search for it in Search reports…).

It shows how much value is still unspent on your gift cards. You can break it down by Gift card status and Currency.

For the money gift cards brought in, open Net sales from gift cards in the same category. To look at individual cards and their codes, use the Gift Cards page in your plugins menu.