"Hate" is a strong word. But "rivalry," "opposition," "fear," "distrust" — these hit the mark closer.
What started all of it? The most direct answer is the Russian Revolution. The overthrowing of the Tsar and subsequent Bolshevik revolution, the Russian withdrawal from World War I, the Bolshevik denouncing every capitalist country (i.e. the USA) as imperialist and colonialist, and dedicating themselves to the overthrow of worldwide capitalism, while the US backed the White opposition in the Russian Civil War (an awful sore spot, that)... the list can go on and on, but the essential take-away is that from the very birth of the Soviet Union, it was ideologically opposed to the USA (even if it marveled its industrial capacity and even made use of many of its methods and experts) and much of the US political establishment (but not all) was ideologically opposed to the USSR and what it stood for.
To the USA, the Bolsheviks represented a dangerous element in the world, one that could mobilize masses in the name of economic equality, but lead to a self-proclaimed dictatorship. To the USSR, the USA represented some of the worst aspects of global capitalism, imperialism, and interventionism. Neither of these are entirely inaccurate assessments.
The periods in which the USA and USSR managed to get along with each other (e.g. World War II) were temporary and unusual, based either in cold, hard realpolitik (such as WWII, where a mutual enemy existed), or a deep fear of mutual destruction (such as the few times in the Cold War when tensions decreased).
I'm in a History of the US Since 1877 class right now. This all goes back to the end of the 19th Century.
Around this time Nativist Americans (people who wanted to stop immigration) had the idea that the people who were immigrating at the time, the 1880s-1920s or so, were inferior to those who immigrated before. The people coming into the country at the time were primarily from Eastern and Southern Europe (Italians, Greeks, people from the Balkans) while those who were already in the US had immigrated from the "nordic" areas like Scandinavia, Britain, France, and Germany.
Living conditions had started to become pretty bad in cities and most new immigrants wound up living in slums. Many of them didn't like this and started becoming a lot more politically active trying to solve the problems. These immigrants also typically began working in the factories in cities which were run by Nativists. The working conditions were pretty terrible with safety being of no concern and most working 12 hour shifts with terrible pay. The immigrants started to try to unionize because of this.
This upset the Nativists, who believed that they were literally superior to the new immigrants and didn't want to lose out on profits paying decent wages and having their employees work decent hours. Some unions tried to strike to get what they wanted, but the companies they worked for stopped it by using scabs, calling in the Pinkertons, or the government would force the strike to end with military intervention. This led to a few gun battles and unions being put down.
Some activists were also members of radical groups and participated in bombings and bank robberies (look at the Sacco and Vanzetti trial for an example)
Animosity and anger had been building over the years with all of this going on. When the Communist Revolution happened in Russia in 1917 the Communist Party started a following in the US as well. The Communists in the US were pretty active in trying to get good jobs in local areas and help cut down on discrimination to an extent. Many people who fell into the immigrant groups previously mentioned began to hear about the Party and what it was doing and thought it seemed like a good thing to be a part of. This also began the first "Red Scare" in the US. Many of the radical and violent acts of the time were committed by those affiliated with the Communist Party.
This whole time the Communist Party was judged by the rest of the nation due to the ideals of Communism. It is pretty much the polar opposite of Capitalism and severely restricts the freedoms of individuals. Therefor the Capitalists in business and government did not like it, nor did much of the American public.
The US Communist party Followed pretty closely with what Moscow was doing, immediately taking the stance of Moscow without consulting members first. This led to some strange behavior with the party. For example, at the beginning of WWII the Party in America was all for going and stomping down the Nazi's because Fascism and Communism are opposing ideals and that's what Moscow wanted. When Stalin and Hitler signed their non-aggression pact in 1939 the US Communist Party took an isolationist stance overnight. Then, when Hitler invaded the USSR anyway the US Party went back on itself again and wanted to go to war immediately.
After WWII the world stage became the US v. the USSR. This is because of the inherent opposition of Communism and Capitalism. This era also saw the rise of McCarthyism. Anyone who was suspected of being Communist in the US was investigated and questioned and many had their lives and businesses ruined. There were a large number of violations of people's rights during these investigations. Senator McCarthy is a good person to look up for more information. He tried to generate a huge commotion politically, accusing almost anyone of being a Communist. He went too far however and lost his seat in the Senate. He died from alcoholism about three years after he failed to win his election.
That's about it that I can remember. Sorry if this is just incomprehensible and sorry about the huge wall of text. This is my first really big post that actually has sources. If some of this is off I apologize.
Sources: For the Record: A Documented History of America, Vol. 2 by David E. Shi and Holly A. Mayer
America: A Narrative History by George Tindall and David Shi
My history professor