Things to avoid:
1. Over-use of technobabble, re-modulation, tachyon bursts, or modified deflector dishes.
2. Warp core breach
3. Earth-like worlds
4. Overly PC or inoffensive attitudes toward sensitive cultural topics. Let's take some risks.
5. Preachy, "holier than thou" garbage about how far humanity has evolved over petty things like "conflict" and "economics"
6. Spatial anomalies of any kind...ever.
7. Continuity bog-down. I have no problem with staying within a general framework, but I could care less about the details. Tell good stories.
8. Large story arcs that concern the nature of intergalactic politics and relations between various governments. Please, enough of that bullshit already. I don't mind a little bit of world-building...but we don't need 4 seasons narrating the complex and unstable relationships between the Breen, Tholians and Gorns in a Post-Nemesis climate where Romulus is weakened. Dull, predictable, male-soap opera crap.
9. No anti-heroes. No. No. No. We have plenty of Breaking Bad, Mad Men, Sopranos etc etc etc etc already. TV is filled with morally ambiguous lead characters...it's old now. Let's figure out how to tell a compelling story without this (now) exhausted trope.
10. No holodecks or re-set buttons that allow you to explore "what if's" in a safe manner.
Things to go with:
1. Complex multi-dimensional characters who have faults, strengths, passions and fears. No one-dimensional characters or relationships. Kirk was a great character. Spock was obviously multi-dimensional. Many of the DS9 characters are good examples as well (Sisko, Kira).
2. Complex arcing storylines about the perils and wonders of SPACE EXPLORATION. Interstellar, The Martian, etc. Let's get out there and TRULY investigate the unknown.
3. Aliens that are truly alien. I don't mind a few humanoid species...but I think it would be fascinating to discover some truly alien forms and cultures.
4. Risky, edgy, fun adventure stories. Again, you can "push the envelope" without losing the fun and light adventure tone of the original series.
5. Chemistry between the leads -real (Kirk, Spock, McCoy or TNG) not forced (ENT, VOY).
6. More of a feeling of isolation and pioneering and less of a feeling of "starship patrol mission of the week"
7. Keep the basics intact: Phasers, Photon Torpedoes, Transporters, Warp Drive, Klingons, etc.
8. Consequences...! I'm not sure that super-complex story arcs like "Game of Thrones" are necessary or desirable, but I'd like to see something more like DS9 where there ARE over-arching plots and that decisions or outcomes are permanent.