Don't want: soap-opera explorations of "relationships". Spock and Uhura are a couple? Fine. But don't make it the central theme of the film. Same goes for Kirk/Marcus, if they go the couple route.
Don't want: lengthy exposition by characters to explain things that should be left for the viewers to work out on their own (for example, if it appears a great distance has been travelled in a short time span, the default assumption should be that nothing important happened in the interim and not "we didn't see every parsec of space that was traversed between the two scenes--why don't they respect speed limits"). I do NOT want some character to then launch into a speech about all the neat things they did in the interim, to explain away the passage of time.
Don't want: any hint, however remote, that this new timeline will be "fixed" in order to merge back to the TOS timeline. Just…no.
Don't want: serious reduction in screen time for Kirk or Spock in order to have a more "ensemble" feel to the film. Kirk and Spock are essential. The other "five" are important but not at the expense of Kirk and Spock.