The answer to your question is: it varied dramatically from group to group.

It's important to keep in mind - Lenin and the communist leadership mostly held national differences as an essentially bourgeois distinction that would eventually disappear under socialism. However, they understood it could not be forced, and Lenin’s goal was to accommodate all minority nationalities until the distinctions between them became irrelevant. This is what led to the establishment of Autonomous Soviet Socialist Republics, or ASSRs, the first of which was created for the Bashkirs in 1917 in order to ensure their loyalty to the Bolsheviks in the ongoing civil war. This was followed by the creation of more - the predecessors to Russia's modern republics, promoting the use of local languages and generally providing greater representation to minority groups (to an extent - "Russian chauvinism" was still very much an issue. If you're interested I can expound on this, particularly in the Baltic states, but since your focus was on the minorities within Russia themselves I'll limit myself to that in this response.)

That being said, not everything was perfect, and ASSRs were still under the whim of the Soviet government. In particular, Stalin was not nearly as friendly to the rights of national minorities as Lenin had been. This reached its epitome during and after WWII, in which Stalin embarked on several ethnically-based deportations to Central Asia and Siberia. The most famous is probably the case of the Volga Germans, but other ethnicities suffered under this policy, notably the Kalmyks, Chechens, Ingushes, Karachay-Balkars, Koreans, and Crimean Tatars. In the case where a group (such as the Kalmyks) had possessed an ASSR before the deportation, that ASSR was abolished and wiped from the record. Most were eventually allowed to return to their homelands following de-Stalinization, but the Crimean Tatars were not allowed to return until the fall of the Soviet Union. What is notable about these deportations is that they were indiscriminate - they deported the entire ethnic group, without regard to the loyalties of the population. The most frequent accusation (as to the reasoning behind deportation) was of Nazi sympathizing, but even soldiers in the Red Army were deported along with their families. In addition, most groups accused largely did not collaborate with the Nazis - and even in cases where there was collaboration, it was greatly exaggerated and used to scapegoat the entire community.

So it really depends on the group. Groups like the Komi or the Volga Tatars, both of which had been under Russian rule for far longer, fared well and were treated relatively equally, enjoying the special status that came with greater autonomy. Other groups - especially those that had come under Russian control much more recently (see: Chechens) suffered greatly.