I know that there were a few prominent Ukrainians, and that Stalin himself was a Georgian (though he was a big supporter of russification, ironically enough). But what about the various Baltic, Caucasian and Central Asian peoples? Were they given equal treatment? Could they migrate from their various SSRs to Russia? And could they rise to important positions within the state?
It was always the case that most party members were Russians, and while all nationalities were represented in party membership, many were underrepresented, such as Lithuanians, Kyrgyz, Uzbeks, Tadzhiks, Turkmen, and Moldavians. Within the individual Soviet republics in Asia, however, the situation improved considerably for local nationalities in the post-war years. In 1924, 8% of Party members in Kazakhstan were Kazakh, but by the 60s it was 40%. In 1967 Uzbekistan , 67% of deputies to the local Supreme Soviet were Uzbek.
As far as I know, movement within the Soviet Union was typically rather easy except for certain points in its history. The Party put in passport restrictions during the famine of the 1930s to prevent a mass emigration of peasants away from the cultivating areas, which could have made the situation worse. The result was, obviously, quite tragic for many. Central Asians also suffered under these policies, as Kazakhstan may have been hardest hit in the 1932-33 famine. In the same period, many national minorities also were repressed and deported as part of dekulakization and the general paranoia about outsiders in the 1930s. These policies did not endure beyond the Second World War. The trend in the post-war USSR was actually for migration from the highest income areas of the USSR to the lower income areas, in part because of the intense state investment in the poorer areas, wage supplements, and various incentives for skilled laborers of various kinds to help in the development of the poorer areas of the USSR. The trend was, therefore, one of generally equalizing incomes across regions, much as it was in the US at the same time (though through different processes).
Emigration from the USSR remained strictly limited until the 1960s-70s when it was liberalized. There were still some restrictions, though. Most emigres had to renounce their citizenship and their right to return. They had to pay an exit fee, which was more expensive for capitalist countries than other socialist countries (by a factor of 10). University educated emigres had to pay back the state for their education, etc.
About Estonia during Soviet period you can read from here: http://www.estonica.org/en/History/1945-1985_The_Soviet_Period/
If you want the short story then only Communist party and KGB had political power in Soviet Union. I wouldn't say that Ukrainians were ethical minority or Georgians, Soviet Union was an union, with 15 Soviet Socialist Republics. There were waves of russifications were students studied more Russian and official business were made in Russian. But that was mostly to homogenize and simplify the process. Mostly Russification didn't work because national identies were quite strong.
About rising to power: It all was decided in the communist party. If you were an active member of the party and knew some right people, then you could be a high member of local kolkhoz (collective farm), whose leaders drove in Volgas and lived a nice life. But you had to be an active member of communist party from early youth. Starting in Communist Youth movements like October children, then Pioneeres and Komsomol
Early after war, when you opposed the Communist party, you were deported or executed. Mass deportations were carried out in 1941 and 1949, in Estonia. First party that wasn't communist socialist party was Eesti Rahvuslik Sõltumatuse Partei (Estonian Nationalist Independent Party) and it was founded in 1988. (http://en.wikipedia.org/wiki/Estonian_National_Independence_Party)
This was mostly studied in high school, by me.It's in national curriculum, so everybody studies it. If you want more official sources then I can search them from Internet but they may be in Estonian.
Here's some reading on the material, it is sourced text http://www.answers.com/topic/russification-1
Aleksander Lankov's book "From Stalin to Kim Il Sung: The formation of North Korea 1946-1960" talks a bit about how the Korean minority in the USSR were treated. They were an interesting and unique minority in their experiences. A large number of Korean men served in the Red Army, so that gave them a bit of prestige amongst the other minority groups. They even served as tank commanders, which was a rare privlidge for minorities. Soviet Korean civilians were also employed en masse by the USSR as translators for their agencies in post-WWII Korea (both North and South, until the schism).
Interestingly, during the period of WWII the USSR relocated Koreans from the Russian Far East to areas around modern Kazakhstan. They feared that they would be working as Japanese informers, so their best idea was to simply to move the entire population. The Soviet Koreans settled in that area and became largely collective farmers, with groups of exiles geographically spaced far away from one another in order to keep them from possibly collaborating with the Japanese.