This is very subjective, as I don't even like the season 3 mockery that's prevelent, I like all 3 live action and the 1 and a half animated seasons too.
Having said that, season 2 is just like Nebusj said, this is Star Trek with the fully developed backstory they didn't bother to before the first season. The Federation is there, and it's always been the Federation and all those times they said Earth in the first year were mispronounciations of Federation.

And Like Harvey said, some of the very best episodes were in the second season, but then there were others. My personal pick for one of the worst, of all episodes is in that season. I usually judge the episodes individually. The second season has a bit more fancyness, can't think of a better word, to it, the sets are a little better, the title fonts match the begining titles, little things that they've improved, but the stories also seem like they were needed to fill in that season more than the first season, as if they had x many good stories and needed more filler. Then the third season, I'll admit, they have even for filler stories.
By filler stories, because they can be really good when well done, I mean instead of starting with an idea for a story that you want to tell they start with a deadline that needs to be met. That's very subjective, too. How many great stories were created because they had to be done by a certain date? We won't ever know. It just seems like there's more of the hurry up and get them made type in later seasons than the first.
One last thing I like about the first season more is there are a lot more background people working on the ship and doing things in the corridors. They all seem to be gone from the second season on. Plus, there are less different bridge officers on rotation.