Yes, you will file the taxes with the foreign country first. You will then receive a credit for those taxes paid unless the income. The tax treaties vary from country to country so whether or not you will need to use them and file with a "tax treaty position" will depend on the specifics of that treaty. The credit is somewhat dollar-for-dollar in that it reduces the tax owed on the same income taxed in both countries in that manner.
However, it is likely that some of your income (such as the social security) will not be taxed in the other country. When this happens, it is possible to still owe taxes on your US tax return, even though you've paid taxes to another country. The tax credits also only apply "like to like" - meaning that taxes on your general income and taxes on your passive income create separate credits that can't be shared between the categories of income.
There is no specific dollar amount to the credits you may claim, but they may not exceed the taxes owed on your US tax return (you cannot get a refund based on foreign taxes paid). If you have more credits than you can use in one tax year, they can be carried over to the next tax year.
While TurboTax cannot help you with the foreign tax return itself (with the exception of Canada, in our separate Canada product), it can help you with the foreign tax credits. For your foreign return, we recommend working with someone in your new home country who is knowledgeable about US taxes as well, when possible,
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