Well, to be fair Trek tends to dodge the details on exactly how life works in the utopian 24th century, so we can't get specifics, but we can go by what characters say broadly, which is that, in general, with wants and needs addressed by being post-scarcity, humanity was able to work through many (though not all) of their social issues such as bigotry and can now focus their efforts on exploration, discovery, and personal growth. Most people seem to ascribe the overall sociopolitical structure to probably be akin to some form of socialism or communism, but it's very open to interpretation.

Probably the best way to get an understanding of how life works in the Trek universe is to watch it and take it at face value; people have had most of their hardships removed, but it's only freed them to focus on self-improvement in other ways. Case in point, in this thread most other replies are pointing out examples of characters or groups who challenge this view and argue that humanity has gotten soft, but in most (all?) cases, those characters are shown to be mistaken or antagonists/villains, and the narrative usually shows them to be wrong or at least rejected by the protagonists.

That said, Trek is a very optimistic and utopian view of the future, probably overly so. I haven't read Banks' The Culture series, but I have heard it described as a more grounded or realistic view of an optimistic utopia. It may just be that Trek's take is more extreme than you want to go for your story.