As a disclaimer, I have done a lot of reading on this topic, but all of my books on it are at my house right now, so I am unable to give citations and exact numbers for the most part. I will edit this post to include citations when I can access them.

The Soviets accomplished quite a bit in the early years of the Space Race, but for their leadership the scientific use of rocketry was basically a sideshow to their missile development, a way to demonstrate the superiority of Soviet science and develop better weapons at the same time.

Following World War Two, both the US and USSR began developing missiles using captured German rockets and engineers. The US managed to capture the chief of the German V-2 Rocket program, Wernher von Braun. Between 1945 and 1957, he would rise to become the head of the US Army's Ballistic Missile program. He spent most of this time working on the Redstone medium range ballistic missile. However, von Braun had always wanted to explore space first and foremost, and he lobbied hard for funding to launch people, animals, or even just satellites into space for scientific purposes. These efforts were largely ignored, up until the launch of Sputnik in 1957.

The chief designer of the Soviet Space Program was a man named Sergei Korolev. He is an incredibly interesting figure, but his life story is slightly outside the scope of the question, so suffice to say that he was brilliant and also dreamed of exploring space. He designed the first successful ICBM, the R-7, which in modified form is still used today to launch the Russian Soyuz spacecraft. The Soviets placed a lot of emphasis on their ICBM program as a result of their lack of a large force of nuclear capable bombers with enough range to reach the US mainland; having a weapon that could deliver a multi-megaton warhead to a major US city in 30 minutes and was unstoppable by any existing weapons system was exactly the sort of deterrent to a first strike they needed. As a result, Korolev's work was given very high priority. In contrast, the US Atlas and Titan ICBM programs proceeded slowly. This was partly due to the state of the Air Force at the time. Unlike the Soviets, the US had a large force of nuclear-capable long range bombers under the control of the Strategic Air Command (SAC). This group had a lot of political clout, and generally went to substantial lengths to suppress ICBM development. Instead the Air Force promoted projects like supersonic bomber aircraft, or very long range cruise missiles, which were a more mature technology at the time than rocketry; the "Snark" and "Navajo" cruise missile programs received several times more funding than ICBM development during the mid-50's. Meanwhile, von Braun was working under the Army, which had no role in ICBMs or other long-range weapons systems, so he was stuck building rockets that would have tactical (battlefield) uses, i.e. small ones.

Korolev managed to persuade Soviet leadership that the US was planning to launch a satellite, and proposed doing so first using their new ICBM. They would get to test their new missile, demonstrate its potential to the US, and one-up them scientifically all at once, and they could even do it before the anniversary of the October Revolution. The idea was approved, and Sputnik flew into orbit on an R-7. After Sputnik, Congress shat a collective brick at the fact that the Soviets could hit the US with an unstoppable weapon, and we had no system that could do the same. Rocket research got a lot more funding, and von Braun's team was given the job of launching a satellite. A few years later, they would be folded into NASA, and the first Mercury astronauts would fly into space on Redstone rockets.

In 1961, Kennedy announced that the US would put a man on the moon before 1970, and NASA set to work creating hardware up to this task. In contrast, the Soviets didn't see the point of trying to beat the US at first; in the eyes of their leadership, the space program was a sideshow, a propaganda effort. Putting a couple batteries and a radio in orbit was pretty cheap and easy, but putting a man on the moon would require new rockets with no military application to be built, and a whole lot of preparatory missions to be launched first. So the Soviets kept pursuing relatively cheap "firsts" to show off their superiority, often to the detriment of the space program as a whole. It wasn't until 1964 that the Communist Party allocated funds for a lunar mission. Even then, the efforts were scattered and frequently derailed. The development of Soyuz, intended as the capsule for a lunar mission, was delayed while the Voskhod spacecraft was built. This was basically a Vostok capsule modified to carry 3 people, which they used to perform the first spacewalk in 1965. Voskhod was a technical dead end, pushing the Vostok design past what would now be considered safe limits; for instance, the capsule was so crammed that the crew could not wear spacesuits, and the lack of testing of the airlock system meant that Cosmonaut Alexei Leonov needed to partially depressurize his spacesuit so he could bend his legs enough to reenter the capsule following his spacewalk. In contrast, the concurrent US Gemini program practiced a whole host of maneuvers and procedures needed for a lunar mission, such as docking between two spacecraft, long-duration spaceflight, and the performance of complex tasks in pressure suits during a spacewalk. Voskhod delivered more propaganda victories, but it cost the Soviets time in developing their lunar spacecraft, so that when the first Soyuz mission launched in 1967, Cosmonaut Vladimir Komarov was killed following a parachute failure. Previous tests had suffered repeated issues with the parachutes, but the mission went ahead anyway, largely due to political pressure to beat the US.