This is sort of a surprise. I know they are different series but I would have thought that subsequent "semi-related" series would work pretty well from the start.
you'd be surprised
The Next Generation's first two seasons are notoriously rough and uneven and never either gelling or able to come out of the shadow of the original Trek. There were terrible behind the scenes ructions, a revolving door writers room, cast members quitting or being replaced, etc. Then the third season got a new head writer and things took off from there.
Deep Space Nine wasn't as bad for its first two years but it still took till season 4 before it took off (when they brought in the Klingons).
Voyager gradually improved but was never great though adding the Borg character 7 of 9 helped a bit, but that was a show where some of the cast were pretty vocal about hating the show. Then the was Kate Mulgrew's attitude towards Jeri Ryan.
Enterprise was decent but people generally accept it got better as it went on and its final season was pretty decent, even if the finale wasn't all that special (there's an ERA thread about that ATM)
Discovery was ok, depending on who you ask but season two was a definite improvement, even if it's still riddled with problems delivering a coherent narrative.
of course, people will tell you different seasons where each 'show got good.'