By this I mean, which race is most renowned for art, music, food and other high social standing things? Culture, in other words.

There was a topic about which race is renowned for cuisine not long ago

I would have to say that the Altmer, Bretons, or Imperials are known for it... but I have a feeling that everyone thinks that their culture is the best, just like in our world

Hlaalu culture seems pretty decadent to have a high standard, I also imagine Indoril does too yet more simplistic.

Pretty much. Most any decision here would be arbitrary and based on a preconception of what "culture" should be. Imperial culture, as an example, can be seen more than one way. Some might see them as refined because of their finery and imports from "exotic" places and all of that, while another group might see it as a morally corrupt society that puts money and personal gain above everything else and that the "finery" is just a bi-product of that.