My father is a Swiss citizen living in Switzerland in a state where there is no inheritance tax to be paid by the children. I am an American and Swiss citizen currently residing in Europe but thinking about moving to either California or Florida. If my father were to die while I were living in one of these states, would I have to pay taxes on my inheritance of real estate, money and stocks located in Switzerland? What about on real estate he owns in the US? Many thanks in advance for your help.

In the U.S. there is no tax on inheritances. CA also does not tax inheritances.
Any property you inherit, your cost basis is the fair market value at the date of death. If you sell later on, you may have profit or loss, which you will report on your tax return.
Your U.S. Tax Return: Tax on Inheritances (http://taxipay.blogspot.com/2008/02/tax-on-inheritances.html)