Actually flesh out the military and economic issues of the Federation before the TNG idiots have a chance to get their hands on them. Thus, Starfleet is actually a military organization with a standing mandate for exploration, while scientific staff are attached non-Fleet personnel from the "Federation Academy of Sciences." The Federation is idealist but not pacifist, and is actually prepared to fight wars if it comes to that. Not every incursion is met by catastrophe or blind luck. Warp drive and the Federation's size both get detailed appearances so that no one can screw with them later. At least one episode features an attack on Sol, such that I have an excuse to describe its productive abilities as well as fixed defensive outlays and fleet presence, and have Spock make some comment about Vulcan and Andorra being comparably defended and industrialized.
Obviously you meant to write Andoria, but I loved the mental image of an episode of Star Trek on a planet full of ski-resorts and duty-free shopping.
Which is one thing I'd specifically try to get rid of on designing sets: the idea that each planet is like a theme park with one environment and that's it. Why does Earth have all these varied landscapes and peoples, but Vulcan is one big desert...
Economically, I'd flesh out a mostly post-scarcity society where advanced manufacturing techniques (which will be undefined in the name of incorporating nanotechnology and orbital refinery techniques later on) and automation in service industries allow everyone a decent standard of living but moving beyond that basic standard requires a free state education and willingness to work. Money is still extant beyond that basic standard, which is comfortable but certainly not luxurious in comparison to what someone with even the most basic of incomes can afford. Thus, most people opt for the "study and work" option. Enough of this utopian anti-consumerist crap...
Though it is consumerist, I would say it still is utopian, but, however, not impossible. Thoroughly agree.
As you can see, most of what I would do is aimed at heading off the morons who took over for TNG before they can turn the Federation into some kind of mush-minded ultra-liberal utopia and make it an actual functioning society with real constraints, struggles, and ideals that it doesn't always live up to.
Which makes for more interesting settings. I'd apply that to the opposition as well.
And that brings me to another thing: make the opposition human as well, in their own way. Which would be one of the great mysteries of the setting (borrowed from the Traveller RPG setting).
When we go to the stars, we find them populated mostly by human looking peoples. Big surprise. And while there are several theories why that is, no one has definitive evidence pointing one way or the other. (Keeps the budget relatively low as well, and provides plot points). TNG used that for an episode. It could have been spread over an entire year of episodes...
Languages for the aliens: use language isolates as a basis at first. We have no clue where it came from anyway. And a former Basque lehendakari already had the Vulcan eyebrows, so... (for the rest, can we ask Tolkien? maybe send him a check?)
So first Andorra planet, then Basque language... next, ship design: flying paellas. Scratch that.
That doesn't mean there wouldn't be truly alien races: instead it means that when they appear they can be even more alien.
In regards of the opposition being human: also try to make them a bit diverse in customs and motivations. The "all Klingons are warlike", "all Vulcans are logical", "all Romulans are sneaky bastards" thing sucks.
Continuity. Story arcs. References to previous episodes. All the stuff they "learned" how to do after watching Babylon 5 (well, probably not from there, but B5 did it better, in my admittedly quite biased opinion).
Also borrowed from B5: no robots or cute children. Unless they get killed.
It is the 60s. But, and that's a big but, I'd put women in positions of authority. In Science and Engineering first. And in other places as well in the opposition. Keep relationships professional, but drop hints that there might be more than that. And use that as a plot point.
Keep time travel nigh to impossible (and do try to avoid paradoxes). Dimension hopping (but something a bit more elaborate than Mirror, Mirror) is a possible alternative to it (or even in combination with it), but still make it next to impossible. Otherwise the suspension of disbelief does not happen that well.
EDIT:
Aargh.
And as I was dreading, just as I was finishing my post, Thande goes and comments exactly on the points I wanted to, and in much better detail.
That's it. I'm going to bed.