I'm not very good at discerning themes from books, but one of the ones that I picked up from Foundation went something like this: "a culture that stops learning is doomed to fail."
What major themes exist in Foundation?
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The major theme I've always picked up from the original trilogy is:
"The flow of Humanity is too large to be consciously changed by any (normal) single individual or group. Instead, it is large social movements that shape history."
You can see it in the early days of the Foundation, where they manipulated the social structure around them (with the help of Seldon's Plan) to gain the upper ground on their more militarily and economically superior neighbors. You can also see it in the failure of General Bel Riose in defeating the Foundation, and ultimately succumbing to the deteriorating politics of the Empire. It took psychic (the Mule) to break the carefully orchestrated Seldon Plan, and it took a whole nation of psychics (the Second Foundation) to put it back on track.
The prequels are about Seldon coming to term with this theme, and realizing that he alone can't change the future. Not even his Psychohistory is enough. He needs individuals who can influence masses without themselves being influenced. Thus his creation of the Second Foundation.
The sequels add another theme:
"Human society is fundamentally flawed, and ultimately will be the cause of its own demise. For Humanity to survive they must change their very nature."
Daneel's master plan of creating "Galaxia" as galaxy wide version of "Gaia" is indicative of this.
As alluded to by Sean's commentg to the first answer, another theme is one taken from Tolstoy's War adn Piece - the tension between the pressure of historical forces of the masses and effects of strong individuals. The former is very thoroughly covered by SystemDown's answer, but while he mentions Mule as a mere "example that reinforces the concept", I view it as introducing this second competing dynamic (ala Napoleon for Tolstoy) of a strong brilliant individual who can try and shift the historical forces.
The theme that has stuck with me, was that it was not possible to stop the collapse, but that there would be a second foundation. The only thing that Seldon could do was to try to ensure that the dark ages between the foundations was as short as possible.
For a non-fiction look at this idea see The Collapse of Complex Societies
I think Foundation is about Asimov questioning what a human ideal future society would look like. He questions what it means to be human - he was from an early age entranced (as I have been) by robotics and how this affects what it means to be human. If a mechanical being looks like a human and thinks like a human, is it human? This is the same theme explored by Philip K. Dick in Do Androids Dream of Electric Sheep? It is less about explaining his opinion on this matter and more about opening up the debate for his readers. And this is what I have always loved about Asimov (which I have been reading since I was a little girl) - the desire to spark a debate. What constitutes a human being? What would an ideal human society look like? Does Gaia represent an ideal or a nightmare? I think it is no coincidence that he began this epic adventure during the war when prejudices about race were in the forefront of everyone's minds. I think it is still an intensely relevant story even today, perhaps even more so today and remains for me the most exciting vision of the future.
"Violence is the last refuge of the incompetent."
Another is that the empirical practice of science and the arts is more important than the academic study of science and the arts - that in certain ways, you have to DO to KNOW. See eg the scene in "The Encyclopedists" where a "scholar" who thinks that meta analysis of secondary scholarship is what is necessary to develop and maintain expertise, and disdains going into the field: Asimov sets him up for ridicule.