Iraq's main commercial airports are Baghdad International, Basra International and Erbil International (all named for the cities they are found in). Regardless of your airport of departure, flights out tend to route you through a handful of major Middle Eastern cities such as Dubai, Cairo, Damascus or Istanbul before proceeding out to other points in the world.
The cities in which these airports are located are considered among the more safe places for United States citizens and other Western visitors, though that is a relative term. It's very important to check the Department of State's most recent travel warnings before making plans.
Some flights out of Iraq will go through Tehran. Americans are allowed to visit Iran, but there are several potential complications with this route. You generally don't need a transit visa if you are not leaving the airport, but you should check ahead to ensure this is still the case as Iran's geopolitical status is in flux. Try to have a printed boarding pass, as e-tickets on your phone may require some extra steps. Also, be aware that entry to Iran can be refused to anyone who has a stamp from Israel in their passport. Israel has recently started issuing their stamps on a separate piece of paper due to this, but older stamps can still cause trouble.
If you're looking for ideas for a trip from Iraq, you can get flights directly to just about all of the regional tourist hotspots and resort cities: Dubai, Sharm El Sheikh, and so on. If you want something a little more unusual, an interesting possibility is Iran's Kish Island. This island is home to Kish Airport and is part of Iran's free trade zone. It is the only place in the country that you do not need a visa to visit; you are granted a 14-day permit upon arrival at the airport and showing that you have hotel reservations or a place to stay. Unlike other parts of Iran, Americans are not required to have a local guide with them at all times on Kish Island. Keep in mind that though it is a little less restrictive than other parts of Iran, certain cultural elements remain, such as the prohibition on alcohol and a requirement for women to at least wear a loose headscarf while in public.
If you're staying at a hotel, inquire if the hotel provides shuttle service to the nearest airport. If it does not, it is highly recommended to get a taxi for safety, even if public transport is available to the airport. Look for shared taxis, which are numerous and stick exclusively to safe areas.
Iraq's airports tend to be on the basic side, but do have services such as WiFi, duty-free shops and baby changing stations. The Basra and Baghdad airports have VIP lounges. Baghdad's airport also has a hotel on site (with a bar and buffet), but be aware that it charges much more than just about any other option in the area.
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