On official level USSR was very egalitarian, anti-racist etc. The same Politburo consisted of people of many nationalities and Stalin was georgian after all

On the other hand common xenophobia was... well... common. Still I'd say nowdays Russia is much more racist then USSR ever was.
Like one of the above posters said, crucial was a mastery of Russian language. If you spoke well you'd be accepted in most educated circles starting from school regardless of your nationality. That is, unless you were a Jew. There was a strong prejudice against Jews unofficially encouraged, officially hushed up. Even then there were plenty of Jews around doing pretty well, though it was much easier with a Russian surname.
It also depended on how mixed your immediate communication circle was. When a few nationalities were mixed together there usually a certain level of tolerance emerged. I've no doubt, it was not always the case, but I lived in a few very mixed communities and never witnessed any serious conflicts based on ethnicity.
Only about half of our class in middle and high school was Russian (and not all officially defined as "Russian" were this, a lot of people with mixed parentage were written as Russians). It was far from common situation, but in large cities and in special cases like the army and newly built towns could happen fairly often.
The one of the Armenians was bullied quite seriously exactly because his Russian was very poor (and he was overweight), though not explicitly for this reason but because he was "a dumb
churka" (I am not sure that "blacks" or "black-ass" was ever applied to any non-white or non-russian back then though it is very common now). Two other Armenians were generally liked despite one being genuinely a bit dumb (but big and strong), both spoke Russian very well. None other non-Russian kids had any problem because of their nationality, but all of them had good Russian. There was even one genuinely black boy (his father was from Burkina-Faso), though that was exotic.
What was discriminating is that all non-Russian speakers were expected to learn Russian or excluded from any possible advancement and good education. Russian themselves had not to learn anything even when living in -SSRs. They had obligatory native language classes there but you could easily fail them and suffer no worse then bad mark in your diploma.