Nikki Haley, the US ambassador to the United Nations, told reporters in a preview of Trump's visit that the president's focus "will be very much on the United States", its role and the relations it wants to build.
The United States was stronger, richer and safer, Mr Trump insisted.
Trump expressed gratitude to Moon for his "dedicated efforts" to reach what he called "a historic milestone in trade". Speaking just before Trump arrived at the podium, Guterres lamented the rise of populism and a fraying of worldwide cooperation - trends the US president has fueled.
"According to the mullahs in Tehran, we are 'the Great Satan, ' lord of the underworld, master of the raging inferno", Bolton said in remarks prepared for delivery at a NY meeting convened to oppose Iran's nuclear ambitions.
Remaining members in the nuclear deal - the UK, China, France, Germany and Russian Federation - say they will set up a new payment system to maintain business with Iran and bypass U.S. sanctions.
In his speech at the opening of the assembly, UN Secretary General Antonio Guterres said trust in the rules-based global order and among states was "at a breaking point" and worldwide cooperation becoming more hard, without specifically mentioning Trump.
Bolton will say in the speech: "The days of impunity for Tehran and its enablers are over".
"In less than two years my administration has accomplished more than nearly any administration in the history of our country", said Trump.
USA officials defended Trump's strategy of engagement with the erstwhile pariah state as the president embarked on a week of meetings with world leaders.
Over the summer, Mr Trump had said he would meet with Mr Rouhani without preconditions to negotiate a new deal, an offer reiterated on Sunday (Sept 23) by US Secretary of State Mike Pompeo and extended to Iran's supreme leader, Ayatollah Khamenei.
The US President is due to deliver a speech in NY on Tuesday morning, during which he is likely to touch on the Iran nuclear deal.
Mr Trump said he had asked Secretary of State Mike Pompeo to take a "hard look" at U.S. foreign assistance, saying that the USA is the world's largest donor of foreign aid "but few give anything to us".
The U.S.is now in the process of reimposing sanctions on Tehran.
"I didn't expect that reaction, but that's okay", Trump said.
Mr Trump also boasted that his administration had accomplished "more than nearly any" other in United States history.
Several African heads of state and delegations have made their way to NY in the United States America, where they will attend the 73rd United Nations General Assembly (UNGA).
Tensions with North Korea have eased since last year's General Assembly, when Trump called North Korean leader Kim Jong Un "rocket man" and threatened to "totally destroy" the country if it threatened the USA or its allies.
Trump said he looks forward to having a "great relationship" with Iran but "Iran has to change its tune".
"(Those lawmakers believe) the nuclear deal has not produced the results and positive developments that were expected, in terms of (Iran's) support of terrorism and recycling of money for terrorist activities, and first and foremost, (in terms of) economic transparency (in Iran) and a clear development of Iranian society", Terzi said.
In his own speech, Iranian President Hassan Rouhani slammed the economic sanctions laid on his nation by the US.
"The North Korean model can not be a correct model, because we can not draw such comparisons", Rouhani, who is to make his own address to the General Assembly on Tuesday, told the NBC network.
Trump however robustly attacked the "globalist" view of the world and vowed that "America will never apologize for protecting its citizens".
He added Mr Trump is inviting others nations who entered the deal in 2015 to trash the agreement, resorting even to threats to achieve the complete isolation of Iran.