I've seen every episode of every Star Trek series as well. Voyager is a bit different because of the premise, they're not on a mission, they're stranded 70 years away from home and trying to get back to earth.
The general theme of Star Trek is that each alien civilisation represents an archetype of humanity that is ostensibly negative, and it's portrayed as self-evident that the Federation is superior. Throughout each episode we see the crew of the Enterprise try to judge these other civilisations by the standard of how the Federation defines right and wrong behaviour, and inevitably the aliens relent, admit their inferiority to humanity and acquiesce to human standards.
When Roddenberry was pitching Star Trek in the mid-1960s, he compared it to manifest destiny in space, and that's essentially what it. The always superior and ever-expanding Federation encounters inferior natives, who — after a short period of resistance — relent and gladly emulate the civilised humans. This notion of the inherent superiority and 'choseness' of humans manifests itself in other ways, too. Like how, for some reason, omnipotent and all-powerful beings like Q regard humanity out of all other species as special, or how smarter and more capable manifestations of artificial intelligence like Data and the Doctor yearn to be human.
The 'soft' white man's burden sort of imperialism of the Federation that regards it's colonial subjects as if it's doing them some sort of favour (representing the United States and the late British Empire) is contrasted with the outwardly brutal and malignant imperialism of the Romulans (representing Nazi Germany), the brutal, senseless violence and 'backward' honour culture of the Klingons (representing the 'Mongol Horde'/Oriental Despot trope) or the cold, uncaring, society-destroying assimilation of the Borg (representing the Soviet Union), but it's imperialism and imbued with a sense of intrinsic supremacism, nonetheless.
As for the Prime Director, we see it used on the show in a way that isn't dissimilar to the way that international law is used in real life. By that, I mean that it's invoked selectively, based on the political needs of the Federation.