Modified from an earlier answer of mine

As popular as the narrative of the "race to the moon" is in popular culture and discourse, both the Soviet and US leadership balked at the escalating costs and dubious rewards from manned spaceflight.

The Eisenhower administration was cool towards space exploration and behind closed doors, JFK was more skeptical than his public rhetoric suggested. JFK's space policy alternated between bellicosity over the Apollo program and a more measured response offering cooperation with the Soviets for a joint lunar mission. NASA had already received a report in 1963 from British astrophysicist Sir Bernard Lovell that the Soviets had no immediate plan for a lunar landing and although JFK brushed the Lovell report aside, his later 1963 UN speech floated the idea of an international lunar mission.

For their part, the Soviet leadership was initially nonplussed by Korolev's satellite program. Korolev's team had gotten approval from Malenkov for development of an artificial satellite in 1954, but the regime's leadership was hardly enthusiastic about the possibilities offered by space exploration. When Korolev and others involved broached the topic to Khrushchev when the latter toured the R-7 rocket facilities in February 1956, the Soviet premier was not terribly impressed by the potential of using the R-7 ICBM for peaceful purposes. In short, the Soviet leadership primarily saw rockets as military weapons first to the exclusion of other possibilities. Korolev and other scientists involved in the development of Sputnik had to constantly lobby their superiors by emphasizing the role satellites could play in defense and that the development of satellites would not hinder the development of missiles for the strategic rocket forces.

The US's soft commitment to launching a satellite and the successful launch of a Jupiter-C rocket in September 1956 convinced Korolev that the Soviet's satellite program was behind the US. The result was that the Soviets scaled back their ambitious satellite, Object D, into something much more simpler and lighter, the PS-1. As before, the design team had to sell the program to the Soviet leadership emphasizing the American threat. Korolev's letter to the Special Committee noted

In September 1956, the U.S.A. attempted to launch a three-stage missile with a satellite from Patrick Base in the state of Florida which was kept secret, The Americans failed to launch the satellite ... and the payload flew about 3,000 miles or approximately 4,800 kilometers. This flight was then publicized in the press as a national record. They emphasized that U.S. rockets can fly higher and farther than all the rockets in the world, including Soviet rockets. From separate printed reports, it is known that the U.S. A. is preparing in the nearest months a new attempt to launch an artificial Earth satellite and is willing to pay any price to achieve this priority.

In reality, the US satellite program was pretty moribund and Korolev mistakenly believed the Jupiter-C launch was an attempt to launch a satellite. But the appeal that a successful US launch would publicly undercut the proclamations of Soviet military superiority and the simplified nature of the new satellite designs meant that Korolev's appeal for an accelerated satellite program was successful and the USSR Council of Ministers signed a decree in 15 February 1957 to launch a Soviet satellite.

The initial Soviet press reports on Sputnik's successful orbit were very low-key. The Soviet's news agency TASS's press release on PS-1 emphasized the boilerplate tropes that this development heralded the triumph of Soviet science without divulging much on the satellite's technical details. The article on the PS-1 in Pravda was not even the top headline for that day's news.

What changed this lukewarm Soviet attitude towards spaceflight was the US's public panic over Sputnik and the global reaction to this advance in Soviet science. Western press reports quickly filtered back to the Soviet leadership and they finally realized the immense propaganda value that space exploration held for the wider Cold War. The Soviet press suddenly started a massive publicity campaign further explaining this triumph and Khrushchev invited Korolev into a personal meeting to discuss the future of the Soviet space program. Eight days after Sputnik's launch, Korolev received authorization to launch a satellite coinciding with the fortieth anniversary of the October Revolution the following month. Sputnik-2, containing a biological element, the ill-fated pooch Laika, launched on 2 November 1957. The rushed nature of the design contributed to the failure of the capsule to protect Laika and the nature of her death was a state secret until 1999. Yet, the propaganda coup reaped by space exploration could not be denied, and portraits of the Soviet space dogs became ubiquitous within the Soviet Union and Eastern bloc. Sending a man into orbit became the next logical step.

The manned Soviet program was in many respects an extension of their earlier satellite and space dog programs. The state would trumpet Soviet space triumphs after their completion and other aspects of the program were kept in relative secrecy. Dual-use or easily adapted technology remained the norm. The Vostok space capsule emerged out of the OD-2 program for a spy satellite and Soviet design teams often stressed the interchangeability of "biological" payloads like humans for those of a more military nature. Electronics components such as guidance and control devices for the space program had to pass through military procurement channels which had a layer of state security to them making innovation difficult. The first batch of Cosmonauts had a carefully groomed media profile and were to be the whole public face of the Soviet space program while the the rest of the effort was kept in relative secrecy. This allowed for sudden public relations coups, but also contributed to grumbling both among the engineers and Cosmonauts that they were pawns in a larger games both against the Americans and the entrenched bureaucracies of the Soviet system. But for all these problems, the Soviet space program's combination of pragmatic mixture of using proven technology and taking calculated risks ( the safety protocols for Cosmonauts were quite bad even by the frontier standards of the time) managed to accrue a steady stream of successes.

Even the first man around the moon was a russian (if memory serves right).

That is incorrect. The USSR had a plan to launch two cosmonauts (Alexei Leonov and Valery Bykovsky) on a circumlunar flight in what was essentially a stripped down Soyuz module launched on a Proton rocket. Their unmanned test missions for that flight kept on failing though. Zond 5 managed to fly a few animals around the moon in Sept. 1968, but it had some issues with re-entry that would have been bad for a human passenger. Zond 6 killed its animal occupants and then crashed Kazakhstan in November 1968. A month after that the US flew Apollo 8, so any PR benefits of a Soviet circumlunar flight evaporated.

Re: how far behind were they, the USSR never successfully launched an N-1 booster, which would have been their equivalent to the Saturn V. There were a number of reasons for this, but three years after Apollo 11 the USSR had tried to launch four N-1s and all of the attempts were failures. At that point they threw in the towel. As a huge space exploration fanboy it pains me to say this, but putting a man on the moon was all about bragging rights. A manned lunar landing 5+ years after the United States did it wouldn't do anything for the USSR's domestic or international prestige. Instead the Soviets focused on Earth-orbiting space stations and tried to act as if that had been their main goal all along.

In 1959, the USSR sent the first satellite (Luna 3) around the moon (and photographing the far or "dark" side for the first time). The USSR never sent anyone around the moon. Although the USSR had many early advances, and the Apollo 1 disaster (1967) resulted in a huge setback for the USA, the latter had advances in large rocket technology that made it possible launch a lunar expedition with the first crew orbiting the moon in late 1968 and the first landing in July 1969.