First of all: DON'T make it a hard reboot. Make it a soft reboot. The adventures of Picard, Kirk, Janeway and Archer all happened. But the new show is set in a new, more "realistic" universe. Like, everyone accepts Picard was a famous Captain, maybe even have a cameo by Patrick Steward. But never again show random bumpy-forehead aliens of the week like they had on TNG. Everyone knows the Voyager returned from a voyage from the other end of the galaxy. But the most egregious "magical" technology is replaced with more realistic one.
Havin said that, here are a few things:
1) No site-to-site beaming: Beaming is a fantastic plot device, and one of the cornerstones of Star Trek. But donm't overdo it. Everytime someone beams, at least on one end of the process should be a transporter plate. In the same vain: Don't have mass beaming of hundreds or more people at the same time...
2) No bumpy-forehead aliens: The only one's allowed are the "traditional" ones like Vulcans, Klingons etc. For the rest of the bunch: Have more elaborate make-up. And if you want to tell a story about a planet-of-hats, maybe don't invent an entire new species, just have it be a colony of one of the already known species. Like a human colony under control of a robot god, or a Vulcan one with strict social stigmas (instead of inventing an entire new species for each case that is then forever forgotten)
3) Robots: The original Trek didn't have robots like the original Lost in Space had. TNG either. This was a budget thing back then. If you're going to re-invent Star Trek, have some regular, normal Earth robots omn ship. Not A.I. like Data or anything, just big, helpful machines.
4) All terrain vehicles/Rovers: It's absurd that on Trek the only means of transportation are either a) starship or b) walking. The wheel is the oldest and most revolutionary invention in human history. It's going to be used as long as humans exist. Have a fucking space rover. Not a Nemesis dune buggy. But something that can be used on planets with toxic atmospheres. (-> see: Prometheus, recent Lost in Space reboot)
5) MACOS: Starfleet isn't "the" military, nor should it be. That means the Federation needs to havve a seperate military (as well as a police). You can make Starfleet the "Navy" - aka the thing responsible for ALL space-travel, wars included. Like the coast guard. But all the ground/sea/air fighting should be relegated to actual soldiers. Who shouldn't be stationed "regularly" on a starship, but only for specific military missions.
6) Federation = NATO/UNO hybrid, Starfleet = Earth forces: Make the distinction clear. Starfleet may be the biggest fleet in the Federation (because humans are an aggressive, expansionist little bunch). But other species have their "own" fleets (even if some of the smaller ones are integrated into Starfleet and use the same ship classes).
7) Nobody is perfect: Scrap the whole "humans are perfect" bit, and replace it with "humans try to be perfect". Like, everyone is giving their best. But everyone still has flaws and faults. The main difference is they are aware of them, and try to overcome them, instead of indulging in them. Basically: Humans didn't naturall evolve to be better, but are educated and shaped by society to be better.
8) Action without shooting guns: TOS was a pretty action-heavy show. But how often did you see people fire guns/phasers on it? Or on TNG? Rarely. Star Trek should be an Adventure show with action. Not a "fight the badguy"-action show with wars in it. There are infinite ways of having action in your show without having gunfights: Running away (from monsters, destruction), natural desasters, fistfights, tournaments, climbing, racing, starship action, spacesuit missions, risky space maneouvers, etc. etc. etc .... Star Trek should have a high action quotient. But it shouldnÄt be about good guys shooting bad guys.
9) More grounded tech: Don't make everything magical force fields. Have real airlooks. The holodeck should be treated as a simulation, like a computer programm (that means: operated, not treated as a reality, with people starting over, rebooting it, using ist for training etc.). Make the replicators REAL advanced 3d-printers. Have people on sickbay "spray" new skin on patients with a spray can, instead of shining a magical light on it.