Huge question. A complete answer would need to encompass political, economic, cultural, religious, and any other number of factors.

I'm going to get my sources out of the way first: Kennan's Long Telegram, Gaddis's A New History, Khrushchev's memoirs, Nixon's memoirs, Westad and Leffler's edited Cambridge History of the Cold War. (I'll also be referencing things likely from FRUS, or archives - -but archives are more difficult to access).

So first, containment was never really meant to be "global" in a genuine sense. Although it took him years to finally explain (likely to come up with) what he meant by containment, Kennan and other like-minded individuals meant creating and reinforcing strong points around the Soviet Union (and for a time China). So, until it "fell" China was meant to be the Asian pivot, after it "fell" that responsibility went to Japan.

Take the difference between Truman's Containment and Eisenhower/Dulles's Liberation. Although the Ike-Foster campaign plan was very quickly dumped after Project Solarium showed them the only places liberation might work were places they couldn't risk trying it, the idea of liberation came about because of the "loss" of China, the Korean war, and the Soviet's aggression in the Middle East and Europe. I could go on about this but the basic point I want to make here is that containment wasn't about fighting wars to change governments, it was about avoiding fighting wars in order to keep the communists from doing so.

Now, I think it is worth pointing out that the U.S. was far far more likely to undertake a covert operation in order to attempt regime change than they were to fighting an all out war. Korea and Vietnam were not about changing governments, they were about stabilising the "friendly" non-communist ones. (Both of these are wildly complicated, I'm not doing them any kind of in-depth justice but making a broad general statement).

A significant reason the United States felt compelled to act against the Soviets, and to an extent (and only for certain periods) against the Chinese was that its main two European allies were simply incapable of it with any discernible effectiveness outside of Europe. The end of the Second World War saw the British and French empires slow crumble quickly accelerate. The French were being thrown out of Indochina and Africa, and the British were being thrown out, asked to leave, begged to stay, and generally drained by the entire process. The U.S., and the Eisenhower administration especially, really would have loved it if Britain and France could/would have basically "taken care" of South East Asia, Africa, and parts of the Middle East. But, Britain and France's leaders throughout this period (with a few exceptions which McIntyre goes in to in Wrapping Up), refused to accept that their empires were crumbling, that they either had to change their policies there or lose entirely, and essentially that the world would no longer accept the old style empires. The Suez Crisis was the tipping point here. Before this period, the Eisenhower administration had basically put zero thought into independent African, and Middle East strategies, and its Indochina policies were still very much expectant of French cooperation. After Suez, while it took a good while to get the ball rolling, the U.S. started treating these areas as regions where if America didn't lead, the options were a Soviet takeover, a British/Franco quagmire, or ruinous chaos cause by misguided and childish efforts of backward peoples trying to run their own states by themselves (Please note that this was the perspective of the Eisenhower administration, not my own).

But why did the U.S. believe it needed to get involved in these regions? Why did American leaders believe, almost to the last man and woman, that if America didn't act, the Soviets would encroach?

Part of this was the Soviet leadership's own doing and actually in the very fabric of Russian communism pre-Gorbachev. In this doctrine, Communism was going to and was meant to "take over the world." Certainly there were leaders who took a softer position on this, Khrushchev and Nixon famously agreed on competitive co-existence, but even then, Khrushchev would not abandon the position that someday Communism would be the world system and state.

To simplify greatly, many Americans and American leaders saw communism as not just the polar opposite of the ideal world as America envisioned it (Capitalist democracies), but as the destructive antithesis to that world. So, this wasn't just a bad an ineffective system, it was the living embodiment of the perversion and destruction of all that was "America."

Is that hyperbolic? Absolutely. However, a great deal of Americans and their leaders believed that to a very strong degree.

That's also, I'd argue, why America never directly fought the Soviet Union. It doesn't prove anything from an "idealistic" perspective, if you literally blow up an opposing system. The West's victory in the Cold War was one that saw the early (NSC 68, NSC 162/2) almost eerily accurately predict the outcome - the West thrived while the Soviet system became a decrepit, corrupt, and self-cannibalising wreck that collapsed in on itself.

The simplest answer is that a combination of realism and idealism drove the U.S.'s actions - it had both a strategic/prudent need to make the moves it did, and considered it part of the American "mission" to oppose communism.

In short, the United States believed in a policy of "containment." Containment was a strategy to stop the export of the Communist revolution due to a fear that socialist states would gradually drift towards the Soviet Union's sphere of influence. Since the Soviet Union had no problem with infringing upon the national sovereignty of other states, the United States likewise had no problem with practicing containment. This is what created the Cold War, in which the United States was nominally fighting the Soviet Union through proxy wars be it in Vietnam or in Afghanistan.

An accessible book that can tell you more about this subject is Rise to Globalism: American Foreign Policy Since 1938 by Stephen E. Ambrose. Ambrose is an easily digestible author -- questions regarding his historiographic issues aside -- and this book should fill you in on any questions you have about the United States and its relationship to Communism. It's a short but good read.

Wouldn't fighting wars to change the systems of government in other countries be considered infringement of national sovereignty? ... why was it the overwhelming opinion in the US that it was the duty and right of the United States to fight it globally?

The United States operated, in the words of one historian, as an 'Empire by Invitation'. In the two decades after World War II, American influence was preponderant, but its influence was desired in many places and even popular with the public. Economic and military aid were desperately needed in Europe. The situation in eastern Europe and Germany made many desire US military alliances or troops. Similar trends occur in Asia and South America. Now, even in the cases when the United States promoted coups in foreign countries, remember that they were invited by army factions, rightist politicians, etc that wanted support for their agenda.

As far as conviction, the United States alone had the capability to take on the responsibility. As Odd Arne Westad argues, 1945-1965 was the only time when there was a true global hegemon. Prior to ~1947, the world was divided into different colonial empires managed from their respective metropoles in Paris, London, Amsterdam. When decolonization began, there was a rapid expansion in the number of nations that could all go toward the Soviet or 'Free World' (Washington's term at the time). If dozens of these new countries fell under the USSR, it would have grave political and economic consequences. From Truman onward, there was a conviction that it was their responsibility, and that only they were capable in the face of the Soviet threat, to stabilize the global order and prevent the rise of communist dictatorships. That meant helping all friendly countries and allies to resist communist opponents. This was their vision of the world at the time.

See two articles by Geir Lundstad, “'Empire by Invitation' in the American Century" & "Empire by Invitation? The United States and Western Europe, 1945-1952".

I will answer part of this question. Most others have dealt with Post WW2 American point of view. I want to give you some of the background required.

I understand that communism is considered largely in the U.S. to be an ineffective and generally bad political system (and I agree)

Im assume by 'communism' you mean a socialist planned economy.

The planned economy was not considered ineffective and generally bad. The economic profession was to a large degree in favor with many of these kinds of ideas. To be sure, they mostly wanted a democratic system, but the principle economics of the idea was in very good standing.

Leading economics textbooks were predicting that the Soviets would outgrow the US in 20 years.

but why was the U.S. so aggressive in its policy of global containment?

The main problem with Communist Idology was that it was firmly against Capitalism had Marxist Communist idiology holds that Communism will lead to a world wide revolution. It is basically a law of history that capitalism will be replaced. So from the very outset, this was a clear problem.

The USSR was from the start a huge problem for 'the West' and even very shortly before the WW2 many people saw the USSR as the major enemy, not Facism or Nazism. When Stalin reformed the country and by WW2 was ready fight germany and beeing able to produce on scales larger then Britain or Germany, it became a large problem in the post war.

You have a country capable of making war on massiv scale (and remember, it was generally assumed that they would grow stronger with time), who idilogically was clearly against all capitalists. This is the needed background.