Even if you are living permanently abroad, you still retain your American citizenship, your American passport and your voting rights. When it comes to the latter, both Republicans and Democrats are glad to help you vote for their candidates in national elections – visit http://www.RepublicansAbroad.org or http//www.DemocratsAbroad.org, and you’ll find extensive information on how to submit your ballot from wherever you are. You’ll also still be eligible to vote in state elections if you maintain an address there.
Many expats elect to apply for citizenship and passports in their new country as well, if they’re eligible. The US does not forbid dual citizenship, but certainly does not encourage it – you only need to make sure that the country you’re in allows for it (for instance, India forbids it, even though they do offer a form of overseas citizenship). Once you obtain your second nationality, the US government may contact you and ask if you meant to relinquish your American citizenship – you only need to answer “No” to keep it, they can’t take it away from you.