Which province's sales tax applies when I ship an order to another province

For GST/HST on goods, what counts is where the goods are delivered, not where your business is. The CRA calls this the place of supply: a sale of goods is generally made in the province where the goods are delivered to, or made available to, the customer. When you hand a parcel to a carrier to take to the customer, the destination you ship it to is what counts.

So a registered seller in Alberta shipping to a customer in Ontario charges Ontario's 13% HST, and the same item shipped to a customer in Alberta carries 5% GST. The full list is in what sales tax rate to charge in each province.

The provincial taxes run on their own rules. BC PST, Saskatchewan PST and Manitoba RST are each set by that province, with its own test for a business outside the province that ships goods in. They are covered on their own pages: BC, Saskatchewan and Manitoba. Quebec's QST is covered in Quebec's GST and QST.

Where it gets specific to you

This rule decides which rate a taxable sale carries. It does not decide whether you charge tax at all, which turns on your registration, and services and digital products follow different place-of-supply rules from physical goods. Where a sale does not fit the plain shipped-parcel case, put it to your accountant. Shopify applies the destination rate once your tax regions are set up (see GST on Shopify sales), and TALISK_HQ records what each order was actually charged.