Question is self explanatory. Correct me if I'm wrong but it is my understanding that the rivalry came from the 1930s/40s whereby the US wanted capitalism and Russia, communism. The rivalry grew from an arms length relationship whereby one another were happy to keep to themselves until when it became obvious that WW2 was going to be an Allied Victory and the division of German land became a question.
Again, correct me if I'm wrong but this is my understanding of it, from my own feelings of how that would make sense I guess.
Relations between the US and Russia were generally amicable in the mid-1800s. The US was "benevolently neutral" during the Crimean War, selling arms and supplies to Russia, and both countries had issues with Britain and France. The US was frustrated with the ambivalent positions of Britain and France during the US Civil War, while Russia was annoyed when Britain and France tried to pressure Russia into granting Poland self-determination during the Polish Revolt of 1863. Russia generally avoided attempts to form a European coalition to mediate the US Civil War, while the US avoided offending Russia on sensitive issues like the Polish revolt.
The most intriguing outcome of this occurred when Russia sent a fleet of six warships to New York in September 1863 in the middle of the US Civil War and in the aftermath of the Polish revolt. Though it did not signal a formal alliance between the countries, the fleet was sent in the spirit of friendship with the US. Russia had several additional reasons to take this action - to avoid getting the entire Baltic Fleet bottled up in case of war with Britain; to get first-hand impressions of ironclad warfare in the US; and to consider purchasing ships from the US. According to George Herring, "for two months, 3,000 Russian visitors attended parades, balls, and dinners, toasting Abraham Lincoln and Alexander II" and the visit "boosted Northern morale and had a negative impact in the South".
The most tangible result of this spirit of friendship between the US and Russia in the 1860s was the sale of Alaska to the US in 1867.
Source: George Herring: From Colony to Superpower, U.S. Foreign Relations since 1776.