I liked season 1 a lot right from the start, though it might be different of course from the older ST series. There were actually enough ST-moments and ST-feeling in the show for me (just take the last episode for example, it's completely based on ST philosophy) and I also found the characters very interesting.
When season 2 started, I thought it's even better for the first five or six episodes. It felt all in all a bit "lighter" and more positive, there was more exploration etc. and each episode, despite being connected to the main arc, told its own story so that you could also regard them, with the main plot in mind, as stand-alone-episodes. This all means that the season fulfilled what they had promised about it in the first place and it brought back the Star Trek elements many viewers missed in season 1 - yeah, but actually only for the first half. Don't get it twisted: I still liked the second half of season 2 and I like the season in its entirety. But for me, something changed in the second half. The episodes didn't feel as episodic anymore as before and seemed to only move forward the main arc. And this one is also a bit of a problem for me: I didn't like the outcome, the big twist about the Red Angel too much. Right from the start, I was so excited about the solution to this riddle, I imagined the Red Angel to be some kind of unknown species, or, even better, some character from the older ST shows, which are chronologically set after DSC. As weird as this might sound, I really had thoughts like "maybe the Red Angel is Picard, we see Patrick Stewart in the very last episode and they establish a connection to the upcoming show". I mean, not that I really thought that exactly this would happen, but I was sure that something or someone really special and surprising is hiding behind the Red Angel and each week I was curious about more details. Well, and then it turned out to be a piece of technology. Hmm...
All in all, for me personally season 2 started extremely strong and as I said, overall I still liked it. But in the end I feel that season 1, despite maybe having less classic ST-feeling in all, was the more "consequent" season. That's why I like it better.