If you are a U.S. citizen temporarily residing outside of the United States, or a member of the Uniformed Services or one of their dependents, you are eligible to vote the entire ballot--including local offices.
If you intend to live outside the United States indefinitely and do not maintain a domicile in your previous state of residence, under federal law you maintain the right to vote for federal offices (President, Vice President, U.S. Senator, and U.S. Representative). That federal law is called The Uniformed and Overseas Citizens Absentee Voting Act (UOCAVA).
There is some variation among states as to whether citizens who intend to live outside the United States indefinitely and do not maintain a domicile in their previous state of residence can also vote in state and local elections. Please contact your local election official directly to find out about the laws in your state.
For questions about voting from overseas and taxation, please see this article.