Officially at least, the date of May 8th, 1961 can be considered the date of conception for the actual Moon landing push. In other words, that is when the inside political backing and lobbying for the lunar mission actually began.

It was on that date that James Webb (then NASA Administrator) and Robert McNamara (then Secretary of Defense) presented a memo titled "Recommendations for our National Space Program: Changes, Policies, Goals", to Vice-President Johnson and JFK, which outlined the reasoning behind why a manned lunar mission was important to the Space Race against the Soviets, and to American global prestige and the appearance of technological/military supremacy.

If you read the portion of the memo I linked to above, you can see the reasoning (even though it is sugar-coated) was to try to beat the Soviets there - plain and simple. In reality, exploration of our nearest celestial neighbor was entirely a distant secondary motive. Science and peaceful exploration was the argument that could be reasonably used to help sell the plan to the public (who had to pay for it), but the USSR had already voiced that goal of a manned lunar landing, and although at the time the USA was in second place in the Space Race in many eyes, America was in it for the long haul, so beating the Soviets to the Moon was achievable.

The end goal was of course not merely to get men to and from the Moon, but also because the technology that would be needed to be developed for that undertaking would trickle down massively, also being of paramount importance to US space-based military and reconnaissance efforts as well. The last line "If we fail to accept this challenge, it may be interpreted as a lack of national vigor and capacity to respond..." really shows that the appreciation of the propaganda value behind getting to the Moon first was clearly understood by the US science and political groups as well. They emphasized that point because they knew politicians have to typically be sold on science, and need to be made to see the propaganda/political value it has before they will agree to fund it.

At this time in 1961, the Soviets were definitely not behind the USA in the Space Race. Quite the opposite really. However, the race to the Moon would require a truly massive longer-duration technological and financial push, and that longer duration timeline (which Kennedy eventually set as 1969 - "before this decade is out...") was one that the United States had a very good chance of outrunning the Soviets on.

To put it very simply, the Russians were committed heavily to the short distance game of low-Earth orbit manned operations, but the USA figured they could beat the Soviets in the marathon to the Moon if they got the full might of American technology, military and public/political financial support.