Like everything else, it's all a metter of opinion and personal taste. It also depends on what you look for in your Trek. The first season had the stronger and more serious plots. The sci-fi quotent was higher and more imaginative. Kirk was more obviously the lead character, going to the planet with the guest star and leaving Spock on the ship. The friendship aspect of their characters didn't really kick in until about midway as they mellowed Spock out some. The supporting cast got more to do in the "personality" scenes and on the whole, it seemed to be a "larger" show for all of that.
Second season saw a definite change in the atmosphere and releationships. Kirk, Spock and Bones were much more "chummy" and there was a greater sense of humor to the episodes. I find many of these early to mid-second season episodes really click. If you had a clunker, you could count on the big three to salvage it, where early first season duds simply hung there, like a hooked puffer fish. There's more action, more "warfare" and I think the "sailing the open seas" connection was stronger. However, Kirk was played more broadly and the background guys got shafted.
The third season (which I enjoy - me and three other guys) really stresses the friendship and brings the supporting characters back into the spotlight. However, there were too many love stories and weird contrievances. Shatner went hogwild at this point, at times chewing scereny like a fat guy at an all you can eat buffet. Spock got actually less interesting as Nimoy played him more stoic, but he was truly annoying in That Which Survives. Bones, apparently, took a bottle of obnoxious pills early on, as the producers seemed to forget the military chain of command aspect of it all. However, the series got sort of spooky this year. The music became more somber and low key, with very little of the Sol Kaplanesque overdrama (much as I love his work, he was insane with the horns). As such, some episodes really gained a layer of darkness, even the turds. If fistfights are you game, this is a great season for you.
Personally, I prefer the feel of the second season, but thought Spock was more interesting in the first 10 or so episodes of the first year. He was much more alien then, not a pointy eared punchline. My favorite episodes of all time is The Doomsday Machine, in the second season. it had the best qualities of the first two years.