If you go to Settings/Taxes, do you have "All prices include tax" enabled? Could you share a screenshot that shows what you're seeing for both the taxes and pricing issue?
Thanks! What you're seeing is normal behaviour. If you have "All prices include tax" then the tax will not appear at checkout as it is assumed the cost of the product(s) already contains tax. What you see on the backend indicates how much tax is applicable for the order.
If you want to charge taxes separately, then you will want to disable the "all prices include tax" option.
Your store may not be set up to charge taxes for specific states or countries. You'll need to review your tax settings in Settings/Taxes to see which regions you charge tax for. If your customers' region does not match the regions you charge tax for then that would explain why taxes do not appear.