Washington's main point is that the Union of the United States--politically, economically, and culturally--ought not to be broken. This thing, this collection of the community into a Whole, is what is most important to the liberties that the United States can offer as long as it remains a republic. Throughout the speech, he pays attention to these key areas of unity, making clear that things like the national morals (which he links indissolubly with a common religion, believing morals to be ultimately derived from religion), and the national identity itself (the idea of being an American, rather than a Republican, a Democrat, a Floridian, etc., etc.) are all-important to induce in the popular mind. That the popular mind must also be "enlightened" (educated) is important beyond measure, for, with the proper context (for this purpose, the supposed American cultural context, but it extends further), that enlightened mind will be able to furnish itself with good government and liberty.
Economically, he is happy to note that all the sections of the United States, existing under equal laws, benefit with trade one from the other on equal terms; that, if this were not the case, not only would squabbles of economy be brought to the fore, but that foreign nations--and treaties with said nations--could result in wars between the states, if not unified. This is classic Federalist talk. The point is that the federal government, by virtue of governing the whole--of governing the individual, not the state--is able to heal the factiousness between existing states by forming them into one country, subject to the same laws, and unified. (As a note, because of this unity, he reckons that the military need not be large, for now the individual countries are merely states in the Union, not subject to war with one another, unless faction arises.)
With all this in mind, Washington makes clear that what he is talking about is an "experiment". The society which he helped build, and which he claims the people built, is nothing more than air if it fails. Hence, he must talk of things to be warned against.
He cautions against disunity of any sort. Party politics is his main aim. That politics, especially state politics, could undermine the unity he establishes quickly. Loyalty not to America, but to Florida, or Arkansas, or Minnesota, etc., etc., whilst disparaging other states allows for faction to grow. He notes that revenge and party-patriotism are common–but that, in addition, such political parties, through their inquietude, tend toward a desire, ultimately, among the people, for security and repose–repose and security that is placed in an absolutely powerful individual, who will use it to seek power and take it. What he seems to mean is that parties are naturally dissident–that, in essence, parties allow for jealousies unneeded, insurrections uncalled for, interference by foreign countries by conniving with the party interest–and, in this sense, tend toward distending the unity of country until the body pops.
He goes far into aiming at factiousness, too. Although, "the basis of our political systems is the right of the people to make and to alter their constitutions of government" one must be aware that said government, the laws under it, etc., are not to be broken unless duly changed. Indeed, the fatal tendency to subvert laws must be avoided, or the forces of faction--seeds of such dissent--will be formed; that, essentially, the minor or majority in favor of themselves and not the common weal, of party and not the common weal, are thrown at the gov’t if such were to happen in this country. The levers of power, needless to say, would be in danger, and the end result could very well be the destruction of the system previously known.
Indeed, to remove factiousness, the delegation of power is itself essential. Speaking as a general principles, the different political elements of the United States gov’t ought to be diffident one to the other, not to encroach one on the other, and even should act as if the others want this to happen–all this allowing a trend toward the checks and balances of gov’t being preserved, preserved as they are essential toward the ability for liberties to exist when power, by nature of itself as power, can be wielded horribly.
Finally, Washington suggests that foreign nations must be loved equally, neither shall there be hatred or infatuation; that, in the first case, conflicts of a passionate nature, illogical, even bloodthirsty (on either side) may become evident, and in the second that the benefits conferred upon the favored nation will be but jealousy-inducing in those less fortunate and the “need” for giving such good things to the favored nation is questionable of itself. In addition, it allows for individuals to subvert the nation-state itself, because, ultimately, they can hide behind love of the favored nation in exclusion of the interests of the nation in which one lives. This all being predicated upon that idea that different nations do not have a common interest, so-called, as they are not under the same gov’t. His opinion, as the whole of the work suggests, presupposes the existence of the nation state--indeed, is wrapped in such ideas. I personally feel his arguments above about the cynical nature of American trade (if it works as he wishes it to work) suggest the system is self-fulfilling. It is realist in the worst sense, because his cautious assumption that all other countries are as cautious as himself, or at least are doing the things which his caution suggests they must do, if followed to their natural conclusion, means that all countries working with any country must also follow along those same cautious lines for fear of dire consequences. A real favor, to him, from another nation, is merely illusory; those nations do not have a real common interest. Hence, all nations must feel the same way as he does, and the separation is impossible to close unless by one big government, but, it seems, the underlying import of his argument is that such a thing doesn't exist or cannot exist.
Modern American society is far from the one envisioned by Washington. His ideas about common unity, his deprecation of party politics, his insistence on a small military, neutrality in foreign affairs (economic and political), the fundamental importance of religious feeling, and even (in some parts of the country) the essential need for an "enlightened" voter base are all lacking.
In modern American society, the importance of a single, unified country has lost meaning. His insistence that, culturally speaking, the identity category of American should be predominant (as opposed to the categories of Republican, Californian, Democrat, etc., etc.) is lost. The importance he places on religious feeling--the common religion of the United States, a basis for the moral feeling which he thinks cannot be disserved from religion--are degrading as time continues.
It is certainly not the "The greatest speech from the greatest President", but it is interesting. I enjoyed reading it, think his ideas on faction are probably just--at least, his ideas on party politics.
One thing of particular note about this speech, and which is sometimes lost in the conversation nowadays, is the importance that was placed by the Founding Fathers on the ability to amend the Constitution. In short, Washington thought this: "The basis of our political systems is the right of the people to make and to alter their constitutions of government. But the Constitution which at any time exists, till changed by an explicit and authentic act of the whole people, is sacredly obligatory upon all."
Why this is important now is not so much because we should venerate the framers of the Constitution--and, of course, we should not: these were people, like ourselves, subject to their own social conditions, fallible, etc., etc.--but because some people do valorize the Founding Fathers. These people go so far as to admit into their reasoning an almost sacredness to these people, suggesting to us that America's laws are laws forever, that change is merely a perversion, whereas--even if Washington's thoughts are expressed with sacred language--the whole idea is that we must follow the laws and change them if we think fit, not that they are commandments to be followed without question.