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You should register your business in any state where your business has a nexus - in your situation - in Florida - and charge and remit sales based on your sale location or customer's address if the order is shipped.
If you are shipping orders to customers in other states - that is their responsibility to pay use tax if applicable.
You may find some additional information here - http://dor.myflorida.com/dor/taxes/sales_tax.html
You may also want to review Business Owner's Guide
Florida DOR provides free tax seminars for business owners - please check the schedule and select the convenient location - https://taxapp2.state.fl.us/gta/taxseminar/
I would need additional information...
Are you perform any business activities in other countries?
What type of activities and in which countries?
Are you selling any goods or services in other countries?
Do you employ workers in other countries? Do you have any business facilities? warehouses?
So why are you concern about responsibilities to other countries?
I do not see any connections...
So far in you are perform business activities in Florida only - you are required to charge and remit to FL DOR sales taxes based on your sale location or customer's address if the order is shipped.