I've often heard it said that the Soviets were ahead in the "space race" and that gradually, the USA reined them in and overtook them, but I think the point about "meaningless firsts" belies that. Was there really a race to put the first satellite in orbit; was there really a race to put the first dog/chimp/man in space, or was it actually race to be first to do it properly and safely.
IMO, the USA was ahead in that last race from the get-go, and never relinquished it, although some might argue that they have now.
The only reason the Soviets got out of the gate quicker than the Americans is because they had a big booster, the R-7, flying first. The Americans only had small intermediate range missiles and couldn't match the Soviet's lift capability, thus were limited in the scope of missions they could fly. Once the Atlas and Titan became fully operational the Americans quickly overtook the Soviets, surged ahead, and never looked back. Even then the Atlas and Titan were smaller missiles than the R-7. It wasn't until Saturn that the Americans had the bigger launch vehicle.
The Soviets had an early lead in lift capability, but it didn't have anything to do with them having a better space program. It was born out of the military situation existing at the time. The Americans had better bomb making technology and could produce less massive bombs. They also had bases in Europe to stage their missiles. The Americans also had a bigger and better bomber aircraft fleet. All these things led to different strategies in terms of missile construction. The Soviets had to have a big booster to carry their heavier bombs all the way to America from the Soviet Union, which led to the R-7. The Americans didn't need anything that big. Thus, when the two sides turned their military missiles into space launch vehicles, the Americans trailed. The Saturn was the first large rocket designed and built from the beginning as a space launch vehicle. Everything before that were re-purposed military missiles.