I think people can not like Spock and Uhura as a couple without it being racist.
This. I don't like them together, and it has nothing to do with their races. I just don't think Spock fits well with her, personality-wise. I know this is because of comparing too much with TOS, but I just can't handle the idea of Spock even getting into a relationship, especially at that point.
But I'd say that Spock would experience racism far more than human minorities (being that we don't see racism against black people, for example, in Star Trek. The only time it's ever mentioned,I think, are times like when Sisko doesn't want to go into the holosuite because the program is set in a time when there WAS prejudice against black people. That would suggest there isn't anymore).
But there still is racism/specieism(?) against other races. O'brien doesn't like Cardassians, in TOS no one likes Klingons,etc. Even Sarek is shown to have what could be seen as "racist" moments against his own son.