How do I set multiple different tax rates for a single product?

Hi,

If you create several tax rules for the same tax category with different tax rates for different zones, then yes, the system will switch automatically based on the address of the customer.
Note also that in the HikaShop configuration, you have the setting "Type of address used for tax" where you can select if the tax system must base itself on the billing address or the shipping address.
Because if you're testing by changing the billing address while the shipping address is configured in that setting, then it's normal that you don't see any change.
Note that if several tax rules match for the same situation, and you don't have the cumulative setting activated in them, only the first tax rule found will be taken into account. If the cumulative setting is activated, then all the tax rules will be taken into account and added to the price.
Now, for further help on this, we would need a screenshot of the settings of each of your tax rules and the addresses you're using on your checkout to test this.