How we cite our quotes: (Paragraph.Sentence)
Quote #1
I constantly hoped that it would have been much earlier in my power […] to return to that retirement from which I had been reluctantly drawn […] but mature reflection on the then perplexed and critical posture of our affairs with foreign nations, and the unanimous advice of persons entitled to my confidence, impelled me to abandon the idea. (3.2-3)
An old trope that you see in a lot of gangster movies is "as soon as I think I'm out, they pull me back in!" Washington's version is similar—he wanted to get out of the game, but he just couldn't abandon people in their time of need. It's the heroic sacrifice that you also tend to see in a lot of action movies.
Quote #2
With slight shades of difference, you have the same religion, manners, habits, and political principles. You have in a common cause fought and triumphed together; the independence and liberty you possess are the work of joint counsels, and joint efforts of common dangers, sufferings, and successes. (10.4-5)
Washington tries to stress the unity of the nation a lot in the farewell address, which isn't surprising because some serious divisions had started to show themselves during his presidency. Here, he urges the audience to remember that they're not so different as they might think and they should work together using their similarities rather than let divisions tear them apart.
Quote #3
The very idea of the power and the right of the people to establish government presupposes the duty of every individual to obey the established government.
All obstructions to the execution of the laws, all combinations and associations, under whatever plausible character, with the real design to direct, control, counteract, or awe the regular deliberation and action of the constituted authorities, are destructive of this fundamental principle, and of fatal tendency. (16.8-17.1)
The principle Washington is promoting here is basically following the rules. People create the government, and they should therefore follow it, too. Otherwise, they're destroying the system that they themselves built. In other words, they're ruining it for everyone else who did follow the rules.
Quote #4
[…] remember that time and habit are at least as necessary to fix the true character of governments as of other human institutions […] and remember, especially, that for the efficient management of your common interests, in a country so extensive as ours, a government of as much vigor as is consistent with the perfect security of liberty is indispensable. (19.3)
Basically, Washington is reminding people that patience is a virtue. If they're not 100 percent happy with this brand-new experimental government system, the first of its kind, maybe they should give it a chance before trying to dismantle it.
Quote #5
The spirit of encroachment tends to consolidate the powers of all the departments in one, and thus to create, whatever the form of government, a real despotism. A just estimate of that love of power, and proneness to abuse it, which predominates in the human heart, is sufficient to satisfy us of the truth of this position. The necessity of reciprocal checks in the exercise of political power […] has been evinced by experiments ancient and modern. (26.2-4)
Remember, friends: absolute power corrupts absolutely. That's why it's so necessary to put checks and balances in the government, to fight against the rise of tyranny. Good thing principled men like Washington were around to put those checks and balances in place.
Quote #6
Observe good faith and justice towards all nations; cultivate peace and harmony with all. (31.1)
It's a little "Kumbaya," but since Washington died about 160 years before hippies were even a thing, we'll assume that's not the image he was going for (unless he was just really ahead of his time). Washington's plea for America to avoid foreign entanglements isn't all about politics—there are principles behind it, too. Mainly that the United States should just be good toward everyone as a general practice.