Truthfully, the best first season in the franchise is THE first season in the franchise... TOS.
None of the spinoffs have fired on all cylinders from the jump like TOS did. Having said that, most of them actually succeeded in multiple ways. (Obviously, otherwise we wouldn't have nearly 800 episodes in total...)
TNG - while the writing room chaos can be felt here and the characterizations may seem off, what TNG truly succeeded at was capturing the feel of being in deep space. The lighting, music, and bold choices in stories more than make up for any shortcomings this initial season had. They expanded on this in season 2 by making space feel TRULY dangerous, and I still feel it's the most dangerous feeling season in the franchise.
DS9 - the biggest problem of DS9's first season was multiple episodes were trying too hard to be a TNG spinoff. But all the episodes where DS9 took full advantage of its own premise... those truly shined. The other thing they did right, starting with the pilot, was the characters FELT REAL. They nailed them right off the bat. One other thing that doesn't get enough credit was the station herself... to this day, the Promenade was THE best set ever. It felt real, alive, lived in.
VOYAGER - while this had arguably the best pilot in the franchise, no series abandoned its own premise more quickly than this one. Which led to many problems throughout its run. But one thing this show got right in the first season was to show how weird and alien the Delta Quadrant is. A living nebula, photonic lattice beings in space, a sentient space twisting phenomenon, aliens who steal body parts to live... it was also refreshing to see a whole region of space where technology was not quite as advanced as Starfleet, and were at a disadvantage only due being alone. High concepts were definitely used here, and that should be applauded.
ENTERPRISE - the Temporal Cold War REALLY dragged this show down. The other problem was season 1 seemed to be slow, for lack of a better term. But this show brought back the sense of wonder and awe at space exploration that we haven't truly seen for years. You can feel the joy and awe just ooze out of all the humans when they see a new civilization, large comet, or new Class M planet. It's the first season that felt like it was actually exploring strange new worlds, and the franchise needed to get back to those roots.
DISCOVERY - this season was a mess, but a lot of it can be attributed to the even bigger chaos behind the scenes than even TNG season 1. And I still feel the overreliance on heavy arcs comes at the cost of having meaningful character stories and beats. But they did a good job in making Starfleet feel like a multispecies organization, not to mention having a large amount of ships. And I have to say, the way they handled "Lethe" in explaining WHY there was such a rift between Sarek and Spock justifies having Burnham be his adopted sister.
LD - I can see why some of the humor felt like too much for many, and the almost lightning speed exchange of dialogue can make it hard to follow. But this season gets the joy of just being in Starfleet, through Mariner. We also get some cool looking ships and aliens that a live action show would be unable to do. It does a lot to not only expand the universe, but also pay homage to all that came before. (Personally, I think this is the best first season of the new shows... so far.)
PICARD - being arc heavy REALLY hurt this season. The first three episodes could really have been done in one slightly extended one. But one thing it did well was get us a look at the Federation from the outsiders' perspective, something we haven't seen since DS9. And you can never go wrong with Patrick Stewart's acting. Plus, it did give us a REAL goodbye to Data... not that abomination known as NEMESIS. That alone justifies this series.
PRODIGY - I have only seen the first 5 episodes so far, so I can't fully comment until I have seen the other 5. But from what I've seen so far, this is right behind LD as my favorite. (My wife commented that it says something about the state of the franchise when I find the animated shows better than the live ones.)
I am trying to reserve judgment on SNW until that season is over, but just from what I've heard and the trailers I've seen... I'm going to really enjoy this one.