The Soviet Union became a nuclear state in 1949 with it's first test codenamed "Lighting Joe", the RDS-1 bomb. It was very similar to the Fat Man as Soviet spies at the time had successfully stolen the design.
As a result of using the Fat Man's design, it was a Plutonium core and had a yield of 22kt.
The SU very quickly moved on to Thermonuclear weapons development by their 4th nuclear test in 1953 with the RDS-6 bombs. These weren't "true" fusion devices however, since they obtained most of their yield from fission reactions.
Some design notes:
The RDS-6s used a U-235 fissile core surrounded by alternating layers of fusion fuel (lithium-6 deuteride spiked with tritium), and fusion tamper (natural uranium) inside a high explosive implosion system. The small U-235 fission bomb acted as the trigger (about 40 kt). The total yield was 400 Kt, and 15-20% of the energy was released by fusion, and 90% due directly or indirectly to the fusion reaction.
Lots more here: http://nuclearweaponarchive.org/Russia/Sovwpnprog.html