Like so many aspects of the Soviet Union, the theoretical freedom of movement was a lot better than the practical freedom of movement for much of the period. Want to live in Leningrad? No problem, just apply for a permit! Want to work a prestigious (or really, any) job? Great, permit applications are right over here. Traveling within glorious Soviet Union? Go ahead and apply for this permit and we'll see that it's given to the proper authorities. Of the three, the third (internal travel) was the most leniently enforced-- for the most part, Soviet citizens were able to travel around the country without much problem.
Thus, the long and short of it is that yes, theoretically you could just apply for a permit to get a new job, move to a new city, or travel to a new region and so long as you met some defined prerequisites you ought to be given the paperwork you needed but practically that meant long, murky, bureaucratic processes (wait times for residence permits averaged around 10 months according to the Helsinki Final Act OSCE report of 1975), dealing with the 'hairy hand' (slang for corrupt officials), and obviously nepotism, favoritism, careerism, scarcity, etc. etc. It was not a simple process and one that Human Rights Watch identified as fundamentally unfree as late as 1992-- meaning in reference to the newly formed Russian Federation, which had inherited much of the legal infrastructure of the then-dissolved Soviet Union. [1] Of course, I'm generalizing a bit here-- if you wanted to go work at a metal smelt in Tula it was going to be a heck of a lot easier than trying to get a permit to teach physics at Lomonosov University in Moscow.
I think it's important stop for a second and underline that like so many things in the Soviet Union which theoretically existed for the entirety of the period, the practical situation was improving nearly the entire time-- especially after the Brezhnev era (1964-1982). To a large extent, Yuri Andropov (1982-1984) tried to hold the hard line too but once Gorbachev came to power in 1985 this process of improvement accelerated dramatically (Andropov's successor Konstantin Chernenko was terminally ill and bedridden before he was even put into power and only lived to serve 36 days as General Secretary). All that is to say, the Soviet citizen of 1939 had far less actual freedom of speech, movement, assembly, etc. than the Soviet citizen of 1969 who had far less actual freedom of speech, movement, assembly, etc. than the Soviet citizen of 1989, but nonetheless, that freedom did not resemble the freedom of someone used to the standards of the neoliberal west of the same time periods.
Consider the Soviet constitution of 1936:
Chapter X Fundamental Rights and Duties of Citizens
Article 118
Citizens of the U.S.S.R. have the right to work, that is, are guaranteed the right to employment and payment for their work in accordance With its quantity and quality.
The right to work is ensured by the socialist organization of the national economy, the steady growth of the productive forces of Soviet society, the elimination of the possibility of economic crises, and the abolition of unemployment. [2]
So we have the right to work, but nothing specific about choice. Compare that to the corresponding text in the 1977 version of the Constitution:
Chapter VII The Basic Rights, Freedoms, and Duties of Citizens of the USSR
Article 40
Citizens of the USSR have the right to work (that is, to guaranteed employment and pay in accordance wit the quantity and quality of their work, and not below the state-established minimum), including the right to choose their trade or profession, type of job and work in accordance with their inclinations, abilities, training and education, with due account of the needs of society. [3]
Choice is specifically mentioned and the practical reality was that there was a certain amount of professional mobility, even upward mobility as well, but it was hampered by the machinery of the state. The Immigration and Refuge Board of Canada stated the following based on reports coming out of various Soviet Studies departments in the Anglosphere (many from professors who had studied in the Soviet Union):
According to Country Reports for Human Rights Practices for 1989, freedom of movement in the Soviet Union is not guaranteed by law. However, citizens are generally permitted to move about freely except within certain areas such as the frontier and strategic regions. [...]
Legally, a person has the right to choose where to live in the Soviet Union. In practice, however, this may be restricted due to the availability of housing and employment in the area. [4]
If I might editorialize for a moment, it's a real tragedy that the system deteriorated so fast (which I don't say as an advocate of Communism in any case), because towards the end-- and especially after Gorbachev came to office-- the Soviet Union (or rather, a clique of individuals within the Soviet government) made earnest attempts to live up to the words written in the constitution and create a consensually government nation in the USSR. One is forced to remember the de Touqueville quote slash mantra of Russian history:
The regime that a revolution destroys is almost always better than the one that immediately preceded it, and experience teaches that the most dangerous time for a bad government is usually when it begins to reform. [5]
Per The Soviet Committee of Constitutional Supervision to the Russian Constitutional Court:
In June 1988 the 19th Party Conference of the Communist Party of the Soviet Union recognized the need to develop a Socialist state under the rule of law. [6]
Three years before the entire thing came apart, there were official government bodies responsible for advising the highest court in the land openly writing that the rule of law was a) not in effect and b) ought to be.
I specifically referenced the difference between the 1936 and 1977 constitutions above because the head of the aforementioned Committee of Constitutional Supervision (CCS), Sergei Alekseev, specifically referenced it during the perestroika period as a potential driver for change:
Alekseev pointed out that some provisions of the 1977 Constitution, viewed in the context of the evolving legal infrastructure of perestroika, could serve as starting points for the Committee's work. [7]