Washington's Farewall wasn't given on his departure from office. It was actually a speech to declare that he wouldn't run again for president.
He covered a broad set of topics. Everyone remembers that he said to avoid entangling alliances. However, he started by saying that if he was ever in a position to contribute as president, his moment was passing and he was not just ready to retire, but it was the prudent thing to do. He was exceedingly humble.
He encouraged the States to stay together, and highlighted all the ways in which their union benefits each other.
He recommended against making hasty changes to the Constitution and form of the government. He spoke of the advantages of checks and balances, and warned against the wrongful usurping of power even if it is necessary for a good act (which goes against the example everyone sites of Lincoln suspending habeas corpus in the Civil War).
He warned against political parties, which he felt falsely agitate the people. He was especially concerned about parties emerging on regional lines.
He speaks of the benefits of education: "Promote then, as an object of primary importance, institutions for the general diffusion of knowledge. In proportion as the structure of a government gives force to public opinion, it is essential that public opinion should be enlightened."
He encouraged us to cherish "public credit" but to protect it by using it sparingly.
He warned against wars: "Observe good faith and justice towards all nations; cultivate peace and harmony with all. Religion and morality enjoin this conduct; and can it be, that good policy does not equally enjoin it - It will be worthy of a free, enlightened, and at no distant period, a great nation, to give to mankind the magnanimous and too novel example of a people always guided by an exalted justice and benevolence. Who can doubt that, in the course of time and things, the fruits of such a plan would richly repay any temporary advantages which might be lost by a steady adherence to it?"
And he gets to the point we all remember, about avoiding alliances: "In the execution of such a plan, nothing is more essential than that permanent, inveterate antipathies against particular nations, and passionate attachments for others, should be excluded; and that, in place of them, just and amicable feelings towards all should be cultivated."
But his belief that we should avoid alliances was informed by the fact that in that era, our geography made us untouchable: "Our detached and distant situation invites and enables us to pursue a different course. If we remain one people under an efficient government. the period is not far off when we may defy material injury from external annoyance; when we may take such an attitude as will cause the neutrality we may at any time resolve upon to be scrupulously respected; when belligerent nations, under the impossibility of making acquisitions upon us, will not lightly hazard the giving us provocation; when we may choose peace or war, as our interest, guided by justice, shall counsel."
He spoke longest on maintaining neutrality.
He closed out by apologizing for his errors, and hoping that history will view him kindly. He said he was looking forward to his retirement among "his fellow citizens."