Who should pay the sales tax when a customer buys a product from me on my Shopify store?

The customer will pay the sales tax to you when they pay for their order. You will then need to pay your state/government that exact amount of sales tax. You will typically only need to pay over all of your sales tax from all of your orders once per month.

You can change your Shopify settings so that taxes are not included in your prices. You will still be charging sales tax, but it will be added on top of your normal price. It is your responsibility to pay your own sales tax in each state that you do business.