The original Wagon Train to the Stars series created by Gene Roddenberry in the 1960s. Very dated now, but still fun (depending which episode you watch, at least) and its historical importance in the development of television science fiction cannot be overstated.
War and peaceArguably, the show's most commonly confronted theme was war. The show was largely an expression of Roddenberry's liberal humanist views and no issue was more prominent in the minds of the public at the time than the war in Vietnam.
Of course, the show advocated pacifism. Kirk insisted their weapons (which had a "stun" setting anyway) were purely for self-defense and Spock could easily subdue opponents without harming them thanks to his Vulcan nerve pinch. In practice, the crew didn't always live up to these lofty ideals and Kirk in particular had a tendency to shoot first and ask questions later, mainly in the name of drama. When they fell particuarly short, Sufficiently Advanced Aliens tended to come along and deliver An Aesop, sometimes by putting Humanity on Trial.
The show's pacifist morality was subverted (or at least deviated a bit) in the episode "The City on the Edge of Forever", which suggests that wars sometimes must be fought for the greater good. Nevertheless, the pacifist leader responsible for the trouble is presented as a moral paragon who, in Spock's words, is "right at the wrong time". It may be an open question whether "City" should be considered pro-war or anti-war.
The human conditionAnother common theme was the nature of human existence. The nature of the series and especially the continued presence of Spock allowed the writers to examine Humans Through Alien Eyes. A touch of existentialism came into play whenever an alien would assume human (or at least humanoid) form with emphasis placed on the strange sensations and emotions said aliens were experiencing for the first time. In particular, these aliens would invariably be overcome and confused by the very human need for love. If female, Kirk often elected to help. Due to his half-human heritage, Spock was alluded to be dealing with these issues on a daily basis.
The show's attitude about the uniting force of humanity is perhaps best summed up in the episode "Who Mourns for Adonais?". Guest character Carolyn Palamas considers betraying the crew of the Enterprise to become the lover of the god-like being Apollo. Kirk tells her "We're human. We couldn't escape from each other even if we wanted to. That's how you do it, Lieutenant—by remembering who and what you are. A bit of flesh and blood afloat in a universe without end. And the only thing that's truly yours is the rest of humanity." On that note, the show held that the destruction of life was the ultimate evil, thus tying this into the theme of pacifism.
DiversityThe issue of diversity was rarely addressed, but the multiracial coed crew of the Enterprise spoke for itself and was taken for granted. By far the weakest link in this theme was the very dated portrayal of women, who all too often were depicted as simple-minded weaklings needing guidance from men. Female guest characters in particular tended to be objectified as objects of beauty with their role in the plot almost certainly centering around sex in some way, usually by being the love interest of a male regular.
As the most prominent female character in the series, Uhura was by far portrayed the most progressively (a relative term). Also notable is Edith Keeler, the aforementioned pacifist from "The City on the Edge of Forever", who Kirk becomes attracted to for her ideals rather than her physical appearance.
TechnologyThe original series often posited questions on the role of technology; and its effect on society. This theme would be continued in all subsequent series/movies.
Interestingly, the series did not go the A.I. Is a Crapshoot route OR the Technology Solves All route. Star Trek overall seems to suggest that as with all things, it can be good or bad (just like Tropes!). Perhaps in keeping with the idealistic/pacifist nature of the series, whether or not technology was a blessing or curse seemed to hinge on whether or not it was tempered by actual moral judgement.
The series itself takes place in the 23rd century, by which time, humanity has achieved many technological advancements, like warp speed and the transporter. However, Starfleet has several rules in place to prevent the abuse of technology. This is most obviously seen in the use of its Prime Directive which forbids contact with a less advanced alien culture for fear of the more advanced society (the Federation) overwhelming the less advanced one.
In one episode, a supercomputer was developed whose purpose seemed to be to dispense with the need for a commanding officer to oversee the ship's operations. However, the machine almost immediately initiates a killing spree; an unfortunate side-effect of its over-literal interpretation of its orders. In another episode, "I, Mudd" the crew outsmarts a group of androids using simple lack of logic to defeat them.
And of course, numerous times in the series, Kirk and co. defeat far more advanced enemies through use of old-fashioned means and even, in some cases, well applied Bamboo Technology (the makeshift cannon Kirk made when fighting the Gorn comes to mind).
Overall, Trek's stance seemed to be that technology as a tool was a great thing. Technology as a replacement for guts and instinct and human intelligence; not so much.