The Soviets had the very first intercontinential ballistic missile, the R-7 Semyorka. This rocket was designed for very heavy nuclear warheads and could be used for manned orbital missions too. Cold war and space race led to the decision to start with orbital manned launches and to skip suborbital flights.
According to this list the R-7 was the first russian two stage rocket.
The R-7 and their sucessors were very successful and reliable rockets. With some modifications, these rockets were also used for robotic moon missions. The transport of astronauts to the ISS is still done with this rocket family.
But the R-7 was a liquid fuel rocket using liquid oxygen and rocket petrol. Preparation for launch took many hours, later ICBM rockets used solid fuel or storable hypergolic liquid fuel to be ready for launch in a very short time.