You may take a number of actions which will affect how the world sees you. You can cut aid or military aid to a country. Doing so may affect the way the people and the government of that country view your administration, but will also affect the way domestic voters view you. Voters want less foreign spending, but foreign countries want more spending on aid to themselves.
More powerfully economic countries have influence on similar countries. For example, how Iran views you will affect how Shia leaders around the region will view you. You can weaken countries by creating sabotage campaigns.
In general people do not like to be ruled by those not like them. Sunnies do not want to be ruled by Shia, etc. This may cause disharmony in a nation.
Domestic voters like having foreign allies. Voters want foreign governments to be stable and to support the USA. The amount of foreign nations friendly or hostile to the USA will affect voter support at home. At the same time, if a foreign government is friendly to the USA but the people are hostile (or vice versa) then that country will be less stable.
Democracy may be important to you, but foreign government may be resistant to democratic changes and allies may turn hostile if pushed too hard to implement reforms which will lessen their hold on power.
A free trade agreement will make domestic capitalists happy, increasing domestic support.
If you support a successful rebellion, the new government of that country will be more inclined to view the USA as an ally. If you attempt to suppress a successful rebellion, those rebels will be more hostile to the USA if they reach power.
Invading a country should probably be used as a last resort, however, sending military aid or American troops into a country can help stabilize the country. The government will have more means of suppressing rebellions and other internal conflicts. Be aware that the more unpopular the USA is in a country, the more likely American forces are to be attacked inside that country.
Try to bring stability into the region while supporting friendly governments. Support democracy as best you can, remove hostile elements, and open up as much trade as possible, but remember, above all else you want to have a high domestic rating. Spending too much or putting American lives in danger is a sure fire way to mess that up. Good luck Mr. President.