US citizens and green card holders who earn over $10,000 (or $400 from self-employment) are obliged to file a federal income tax return wherever they live. If they are resident in another country, they have to file a return there too. In both cases, penalties can be harsh for not filing, so it's important to understand both sets of obligations as soon as possible. The first step is to understand what needs doing and by when.
Taxes owed for any calendar year are due by the following April 15th, however the return (form 1040) doesn't need to be filed until June 15th. Furthermore, you can request an extension for filing your return until October 15th online.
The Foreign Account Tax Compliant Act (FATCA) meanwhile requires US expats to attach form 8938 to their return declaring overseas assets of the total value is over $200,000.
Furthermore, US expats who have at least $10,000 dollars in total in accounts outside the US at any time during the tax year must also submit an FBAR (Foreign Bank Account Report) declaring their overseas bank accounts. In practice this is form 114 which from 2016 must be filed online by April 15th, though again an extension will be available until October 15th.
The IRS has two main provisions to help prevent double taxation, and the US also has tax treaties with many countries, so most US expats won't be taxes on the same foreign earned income twice.
The Foreign Earned Income Exclusion (FEIE) relieves US expats from paying tax on the first $100 thousand, more or less, of income earned abroad. The exclusion can be claimed using Form 2555, attached to form 1040. If you are married to a US citizen or green card holder who also has foreign income, you can both claim up to the full amount.
The Foreign Tax Credit meanwhile allows you to claim as a credit against your US liability a dollar for every dollar of income tax you've paid to the government in your country of residence. It can be claimed alongside the FEIE, however depending on your circumstances, it can make more sense to just claim the Foreign Tax Credit. As a rule of thumb, if income taxes are higher in your country of residence than the US this will be the case.
Dates and rates in your country of residence
In most cases, you can find out about your tax obligations in your country of residence with an online search. The first things that you need to know are filing dates and tax rates, along with whether you can file online or whether you need to fill out a paper form. If the latter, can you download it, or do you have to go somewhere to ask for it? In most countries you'll need to register with the national tax authority soon after arriving, so you should find out how this works early on. If you are employed abroad, your employer should be able to help with this. If in doubt, ask someone you trust to recommend you a good local accountant.