I see your post is a couple of months old, so maybe you've made your decision by now. On the chance you haven't, I'll give my perspective as someone who recently upgraded from DVD to Blu-ray.
For the Kelvin timeline movies, yes, absolutely, Blu-ray is the way to go. I recommend the Compendium version of Star Trek and Into Darkness as Into Darkness has the IMAX sequences shown taking up the entire screen, closer to how it appeared in the theater. The sound and picture are great and this version contains the most features.
I'll say I got my original crew and Next Generation crew movies on blu-ray cheap from a popular online auction site. That was the way to go and I don't really regret it. The Motion Picture and The Wrath of Khan look best of the original crew movies. The picture is clearer and sharper and a clear upgrade over dvd. Of course, higher resolution and better sound should always be reason to upgrade. However, as others have noted, the process which was used to upgrade the scans to blu-ray makes some of the skin tones look unnatural. I don't think it's as bothersome as some allude, but it's probably a matter of preference.
In Star Trek III: The Search for Spock, I noticed that Klingon Bird of Prey appeared BRIGHT green on blu-ray. I don't recall it looking that bright on vhs or dvd. It really stood out (to me) on blu-ray. I also noticed some bad special effects during the scene where Kirk as fighting with Kruge on the planet. The higher resolution made some of the bluescreen work really jump out and look cheaper than it's ever looked before.
So, for original crew movies, I'd say Motion Picture and Wrath of Khan are clear upgrades. I had issues with 3. 4, 5 and 6, I was ho-hum. I suppose it was an upgrade, but not a clear standout upgrade.
For the Next Generation movies, I have only watched Generations all the way through. It had higher resolution, and the Enterprise-D looked clearer and crisper than I ever noticed before. I have popped in each of the other movies, but not watched them all the way through (yet). It seemed Nemesis was the clear standout of this set. It really looked good on Blu-ray, a clear improvement, but it is still Star Trek Nemesis.
I did like that the Blu-rays featured some cut scenes and behind the scenes stuff which was not featured on the DVDs. My thoughts, get the Kelvin universe movies on Blu-ray (preferably the Compendium edition). For the original crew and TNG crew, if you can get them cheap enough, you probably won't regret it, but don't expect them to blow you away.
I have not see the Director's Cut of Wrath of Khan on Blu-ray (I did attend the theatrical event in September). So, I don't have a basis of comparison for that one, but I'm curious to check it out. I do not have a clear preferred version of TMP (I like the theatrical cut because it's the shortest (!), but the Director's Cut does add more character to Spock), and I avoid the director's cut of Undiscovered Country at all costs.
I hope our input has proven helpful.