There are lots of things I've found illogical in the Star Trek series; I've followed all of it except "Enterprise". In my later years of following it, however, having to make an appointment with The Tube became a big hassle, so the only CRT's I now watch are computer-controlled ones.
That aside, some of the less-logical parts are halfway-reasonable concessions to Earthbound audiences. These include noisy explosions, stars racing past while traveling through interstellar space, and spaceships never being seen upside-down. One can certainly snicker at a noisy explosion in outer space, but a silent one might have a lot of viewers wondering what happened to the sound.
Some of it was just plain melodramatic, such as consoles spewing sparks and ships being shaken around; however, a crew seemingly unaffected while worriedly contemplating a shield-strength strip-chart display may seem rather dull.
There were some illogical features, like there seemingly being only one Transporter in all of the TOS Enterprise -- one would naturally expect more than one, especially in a spaceship whose habitable parts can split in two.
There was a lot of absurd verbiage that's been aptly named "Treknobabble"; one might expect much shorter and more generic words in many cases.
And the biology -- the idea of a human-alien cross is just absurd. Different biological histories would make a true crossing impossible. Horses and donkeys had diverged about 3 million years ago, and are still very much alike, but crosses between them are sterile. And those are very *close* species.
I can understand using human actors, but making the aliens behave too humanlike is IMO illogical. Though some episodes did make some stabs at showing exotic-behaving ones.