RPGs:
Out of the "official" RPGs each has its strengths and drawbacks.
FASA produced a TOS RPG in the 80s. It actually was kind of considered a source of "canon" by lots of fans back then (until TNG came out and changed it). They had a Klingons book that expanded on John M. Ford's brilliant Klingons. I'm working on creating a "retroclone" of the system. Very 80s and could use some updating.
Last Unicorn Games had a beautiful TNG game in the 90s. It was actually the first RPG that I ever bought. A little complex but character generation was pretty flexible.
Decipher's 2000s RPG was kind of cool. It didn't do things like other games at the time and was not super popular. I liked it a ot because it gave you lots of options. It's best if you have the Starfleet Operations Manual, which offers lots of options beyond Federation characters.
"Semi-Official" RPGs include the various versions of Prime Directive, a game set in Amarillo Design Bureau's Star Fleet Universe. The biggest story strength of these games are the Klingons, in my opinion. Much cooler and kind of like an anti-Federation (though not sworn enemies of the Feds). It was made for GURPS, d20 and an original ruleset (which I haven't read). Mongoose is supposed to release a version for Taveller, but that's been on hiatus for a long time.
As far as unofficial RPGs go, there's Lasers and Feelings, which is fast and easy. I think that Where No Man Has Gone Before is probably the best Star Trek RPG ever made. It has lots of options for creating stories and flavorful character options.
There were a couple other games, one called Star Trek: Adventure Gaming in the Final Frontier, and one called Starfleet Voyages. I have never seen them before and would love to get my hands on a copy.
Board Games
I've never played a board game or a miniatures game set in any of the Star Trek properties. I am actually creating a board game inspired by the old EGA Trek video game (it has no elements of Star Trek in it, though, so I don't get sued). There's a thread about it
here.
Card Games
Decipher's Star Trek CCG is the most famous. It started out fairly focused and became bloated. At the same time it offered a gonzo experience with lots of possibilities. A second edition (which I never played) made it faster and more balanced. The coolest thing they ever made was an 18-card set called The Fajo Collection, which I still have. The cards in that set had the highest production values of any CCG product and were really fun.
Fleer (the baseball card people) produced and underrated card game that had cards made from the same paper and finish as a regular poker deck. Fast play, simple rules. Faded fast and it shouldn't have.