One other thing I would definitely do, no matter what, following BlackWave and DValdron here, would be to hold on to the core concept of optimism in TOS. That's a huge part of what's made the show so appealing for so long - the idea that things will get better for humanity, that we'll get through our current messes on Earth and make it to the stars - and it's something that fans of Trek can agree on no matter their political viewpoints.
I'd keep the primary-color uniform tunics for day-to-day shipboard work, but use field jackets for planetary expeditions (such as you saw in "The Cage"/"Menagerie" and The Wrath of Khan), and fancier dress uniforms that paid more tribute to Starfleet's naval heritage.
I strongly recommend the book Inside Star Trek by Herbert Solow and Robert Justman. Between them, the two men pretty much ran Trek's production side (Solow, a NBC executive, took care of administration and finance, and Justman oversaw day-to-day production), and they give tons of information on the challenges involved in making the show and how they were overcome. Having more non-humanoid aliens is nice, for example, but it's just not going to happen given budgetary realities; even the more elaborate makeup jobs took a lot of money and trouble (the book talks about just how difficult it was to create pointed-ear "appliances" that Leonard Nimoy could wear comfortably). Kitbashing is also not the most optimal solution to the problem of devising new ships for Starfleet, but at that time, it's the most practical and cheapest one, so as head honcho on the show, I'm going to have to strike the same kind of balance Roddenberry & Co. did between what was wanted and what was feasible/affordable.
I'd certainly find a way to fit both of Starfleet's primary missions - defense of the Federation and scientific exploration - into its charter. There's no reason why the primary mission of the Enterprise and her sisters, to quote the famous intro, couldn't still be, in peacetime, "to explore strange new worlds, to seek out new life and new civilizations, to boldly go where no man has gone before", but, when hostile alien polities threatened, to become to keep and defend the peace. It'd be worth emphasizing, if Starfleet's military role was played up more, that it was a defensive force rather than an offensive force - or, to quote Captain Pike in the 2009 reboot, "a humanitarian and peacekeeping armada". I like that; non-aggressive and peace-loving at core, but always ready to repel aggressors with overwhelming force if necessary.