In terms of moving to a new town or settlement, citizens could move, but from the mid-1930s on there was an internal passport system, and citizens needed to be officially registered in their place of residence. This was done specifically to limit immigration from villages to cities: if someone moved to a city, but their internal passport listed a village as their place of residence, they could be denied social services such as education or housing, and sent back to their home village (the hukou system that still operates in China is broadly similar). Getting an apartment in a new town meant putting your name down on a list and receiving one either from local authorities, or from a local employer, and so was very much tied to getting a job (I'm saying "apartment" for the Khrushchev era on. Previous to this there was horrible overcrowding in Soviet cities and inhabitants had to make do with a room or literally a corner). A final note is that for much of the Soviet period there were "closed cities", which were cities involved in particular defense industries, and these cities were absolutely off limits to unauthorized personnel.

More can always be said, but here is a good rundown from u/Minardi-Man how one would look for a job in the USSR. One addendum I'll add is that your employer and your trade union were the distributors of most of your social benefits: housing, social insurance, medical care, housing, and even (depending on the time and place) your hot meals.