This one's easy. What self-respecting Trek fan can avoid the immediate image of Patrick Stewart in the big chair when that word comes up. I could talk about how to engage players, or how to engage as a Game Master, but instead I'm going to muse a bit about Trek RPGs. So assume a standard orbit for a moment and let's talk about playing Star Trek.
Star Trek: The Roleplaying Game (FASA)
"FASA Trek" was my first Trek RPG, and one of the first RPGs I owned. My Mammaw and Pappaw purchased it for me in 1986. Inside the box was the Cadet's Orientation Sourcebook, the Officer's Manual, and the Game Operations Manual. I was immediately in love. In the days before the internet, when all we had was novels and the Franz Josef Star Fleet Technical Manual, FASA Trek was a godsend for fans wanting more. It used a percentile system, so the math was very straightforward. The system itself was a little too wargame-like for Star Trek, with a system of maps and squares and Action Points that could have been much simpler, but that didn't bother us. We had a blast with it. The adventures written for it were generally amazing. The sourcebooks were fountains of great background like the John M. Ford material on the Klingons that I still use in my own Trek games over the Biker Vikings from TNG onward.
The starship combat game that was compatible with this RPG was great, as well- to the point of having "consoles" for each player to move cardstock chits around to do their jobs. Engineers balances power and allocated resources to the helm, weapons, sensors, etc. The players at those stations then took that power and used it to perform their duties. It was like the Artemis bridge simulator years before computers made that possible. I *loved* this version of Star Trek. It was pretty much all we had aside from novels, fanzines and Star Fleet Battles for my whole childhood. We had to fill in the blanks, and these books helped us do that. I love that I've become FB friends with some of the folks who worked on it, especially Blaine Lee Pardoe, who wrote one of my favorite adventure modules (The Strider Incident) *and* the TNG First Year Sourcebook which I mined for all sorts of ideas- like the Bridge Command Specialist. That concept alone is a fun one to play with, and explains why Geordi, as a Lieutenant (j.g.) is left in command of Enterprise when Picard and Riker are down on Minos, rather than any of the more senior officers. I only wish the promised Ground Forces Manual would have been published.
Prime Directive (Task Force Games/Amarillo Design Bureau)
This came out when I was in high school. It's based on the Star Fleet Universe, a setting created when Task Force Games sub-licensed Trek from Franz Josef, the publishers of the original Star Fleet Technical Manual. Think of it as Trek if Trek had been written by Tom Clancy. It's a much more bellicose and martial take on Trek, and there are some limitations to the license- nothing outside the Original Series and Animated Series, limited use of character names, etc. So what you have is TOS Trek where the Cold War with the Klingons went hot, and became The General War. There's even an Axis & Allies from Hell style game called Federation & Empire that allows you to play out this scenario. Prime Directive exists in this version of Trek, and players are assumed to play members of a Prime Team- sort of Star Fleet SEALS, trained for "novel contact" missions, military missions, diplomatic missions, etc. The idea is to have a team of highly trained personnel to handle dangerous or unusual situations aboard your ship so you're not constantly beaming down your top five senior officers and Ensign Ricky. The system was interesting, if not entirely smooth, and I really liked the action-oriented Trek setting.
Modiphius Star Trek
I'm really an old-school gamer, but I have to express a certain appreciation for the Modiphius 2d20 system and the Star Trek game that uses it. It's got a lot of new-fangled game mechanics that futz with the dice- so it's not going to go over well with some of the hardcore Old School crowd, but it lends itself to some amazing storytelling.
I've not gotten to put it through nearly the paces it deserves, but it's been a lot of fun so far, and encourages player creativity in the interpretation of rules, spending of Momentum, etc. I wanna see where this goes.
Starships & Spacemen 2nd Edition
OK, This one is a guilty pleasure. It's an update by Goblinoid Games of an old title from the 70s that was Star Trek with the serial numbers filed off. It really feels like Trek D&D. Classes, levels, saving throws, all that. You can even brorrow D&D monsters with a tiny bit of tinkering. It has a whole resource management system for ship travel and combat, and a lot of other cool "almost Trek" elements. I find it a lot of fun to play on its own merits. I even printed the character sheets in uniform colors.
Other Trek Games
There are other Star Trek RPGs, to be sure. GURPS Prime Directive, the Decipher Star Trek RPG, the Last Unicorn Games RPGs, and I'm sure there's one or two I'm forgetting. But the ones I elaborated on are the ones I have spent the most time playing, and enjoyed for their Trek-goodness. Why choose only one? IDIC tells us that infinite diversity in infinite combination is a good thing. LLAP, folks, grab a Trek RPG and ENGAGE!