It's kinda complicated.

At the beginning (before Lenin's death) Soviet authorities were mostly against renaming because it's quite expensive and, someone can say, useless. The idea of renaming was popular because local leaders wanted to eliminate names with religious connotations or to rename places that were named after their owners, but central authorities rejected the majority of such requests. Actually, from December 1923 it was strictly forbidden to rename any city or village with a post station in it.

Funny fact: some people from village Drischevo (something like, Shitty Place in translation) wanted to rename their village to Leninka in 1923, but they weren't allowed to do that.

As a result of this politics, the number of renamed cities was quite small. Only 27 cities were renamed in Russia in 1917-1924. Petrograd (Saint-Petersbourg before 1914, the city of Peter I) became Leningrad (because of Lenin). Tsarevo-Sanchursk became Sanchurck (part with tsar or czar, king/emperor in Russian, was eliminated here). Svytoy Krest (Holy Cross) became Prikumsk. Ekaterinodar (literal translation: gift from Ekaterina II) became Krasnodar (literal translation: gift from the Red ones). And Simbirsk became Ulyanovsk. The last case is quite unique, because Simbirsk doesn't have any connotations to Russian empire or to Church but Lenin was born there and his real surname was Ulyanov.

All these politics were changed after Lenin's death. It became easier to rename a city. Also, the goal of first renamings was to eliminate reminders of old regime (religion, empire and so on). But the goal of new renamings was different. They were needed to glorify new leaders.

So, Tsaritsyn became Stalingrad. Vladikavkaz became Ordzhonikidze. Vyatka became Kirov.

Sometimes, the proposals were still rejected, and I'm not sure we know the reasons behind these rejections. In 1931 people of Chelyabinsk asked to rename Chelyabinsk to Koba (it's a Stalin's nickname) but Stalin forbade this personally. After that people of Chelyabinsk wanted to rename their city to Kaganovichgrad to glorify Lazar Kaganovich, but this idea was also rejected.

Sometimes, Soviet authorities had to rename the same place several times in a row. For example, Merezhnoe village was renamed to Ezhovo in 1938 (to glorify Nikolay Ezhov, the head of the NKVD at the time), but after Ezhov was fired and arrested (and later executed) Ezhovo became Chkalovo (to glorify famous Russian test pilot).

Village Batalpashinskaya became Batalpashinsk city in 1931 to be renamed in 1934 to Sulimov (to glorify Daniil Sulimov) to be renamed in 1937 (after Sulimov was tried and executed with a little help from Nikolay Ezhov) to Ezhovo-Cherkessk to be renamed (after Nikolay Ezhov was arrested) to Cherkessk (1939).

Oh, it's too long.

Okay. After Stalin's death some cities were renamed again, and Stalingrad became Volgograd. And Stalino became Donetsk.

During the last twenty years of the USSR the renamings were relatively rare. They mostly happened after the death of some Soviet leader. Naberezhnye Chelny became Brezhnev, Izhevsk became Ustinov, Rybinsk became Andropov.

After 1985 some cities were renamed again to eliminate the reminders of their Soviet past and to allow them to go back to their historical roots. Gorky became Nizhny Novgorod. Kuybyshev became Samara. Leningrad became Saint-Petersbourg.

At the same time some cities seem to be okay with their soviet names. Krasnodar is still here. Vyatka is still Kirov and so on. This process was mostly based on locals wishes, so some cities decided that they liked their Soviet names after all.

These later renamings sometimes were quite strange. In Russia there is a tradition to name the area around the big city after this city. So, area around Moscow is called Moscow oblast, and area around Krasnodar is called Krasnodarskiy kray (I don't know what the difference between kray and oblast actually is, but it's not important here). The thing is, in some cases cities were renamed, but areas around them weren't. So, now in Russia there is Saint-Petersbourg, and it's the capital of Leningradskaya oblast. Or, there is Yekaterinburg (another city that was founded by Yekaterina II) but it's the capital of Sverdlovskaya oblast because in Soviet times this city was called Sverdlovsk.

And in almost every Russian city there is a Lenin's street. Often, it's the main street of the city.

Small clarification: when I talked about Soviet renamings I mentioned not only Russian cities and villages, but cities and villages from another Soviet republics as well. The last part of my answer (back to historical roots) is about Russia, because I don't really know what did happen with Soviet names in other countries.