In general, you're looking for these two configurations:

• Accounting -> Configuration --> Accounting -> Fiscal Positions
• Accounting -> Configuration --> Accounting -> Taxes

For Fiscal Positions you want a setup similar to this:

... where AT w/ VAT ID is

... AT w/o VAT ID is

and so on for all Countries relevant (setting a Foreign Tax ID at a Fiscal Position requires you to also specify the Country (of delivery) it applies to).

Whenever you set a new Country, you will be asked whether you want to create the corresponding taxes for that Country - you can do so by clicking here, or manually add the relevant tax rates (might be meaningful since adding all tax rates for all countries could become quite a mess).

Ultimately you should end up with a large catalog of Taxes under Configuration. From here you need to specify which EU country-specific tax rate is replacing your default US tax rate. For example the default Austrian tax rate of 20% would replace the default 5,75% (*) for AT w/o VAT ID, 0% would replace 5,75% (*) for AT w/ VAT ID and so on (specifically note Fiscal Position and Replaces here):

* Sorry, I'm not aware about your default tax rate

Then, when you create invoices for a B2C or B2B customer, the fiscal position and tax rate will be automatically set at the invoice (NOT at the customer):

... and an invoice would look like this:

Note: The exact tax rates, replacements of tax rates, accounts etc. should be discussed with your accountant!