Crysis and Crysis Warhead are must haves for any FPS player. I recommend Far Cry also for historical value. Remember it was released in 2004 when you play it and hopefully you'll get at least some sense of just how incredible it was. Crysis 2 was decent enough as a standalone shooter, but pretty far from a must-have, and really isn't much in line with 1 or Warhead. The nanosuit (in name) and prophet are really the only connections to the original. I haven't played 3 due to origin and the reviews stating single player was extremely short.
Also must-play is the first F.E.A.R., unless you don't like horror games. I liked the expansions for the original in that they just gave more of the original gameplay and atmosphere. Extraction Point has what I think is the best scare sequence in the series. The stories are crappy though, and in fact were removed from canon. Not that matters much anyway, because like Crysis, FEAR 2 and 3 are decent enough shooters (2 a little less so) but pretty much sequels in name only. Come to think of it, I never even did return to 3 and replay as Fettel.
If you like the third person shooter/rpg hybrid type games, Red Faction Guerrilla and Mass Effect are great. I think pretty much the rest of the world would disagree, but I think ME1 is miles better than 2 and 3. It's a little rougher on the edges, and if you don't like repetitive grindyness (I don't mind, and even kindof like it) you'll probably hate the side quests, but I like the gameplay mechanics soooooooooo much more than what they were changed to, and again it's a lot more atmospheric. 2 and 3 I played more for story and completion than gameplay.
The STALKER series is another great FPS/RPG hybrid, although as stated before they are REALLY buggy and pretty poorly performing, requiring mods/user bugfixes to really play right. Once again I find myself saying the original Shadow of Chernobyl is the best for atmosphere and story, but the gap is actually quite small this time to the sequels. There were some pretty neat mechanics added in Clear Sky and Call of Pripyat to make up for it though. And just because it needs to be said again, for the love of god don't try to play these without bug fixes. SoC could be played first time with the Complete mod without changing the core of the game too much and it includes all the bug fixes, but I did find some parts of it undesirable. For example it made nighttime pitch ****ing black and pretty much unplayable without a night vision upgrade. Some of it's changed effects I also didn't like compared to the original, but all in all it stayed pretty well true. If you're not a stickler for "the way it was meant to be" you should probably just skip playing through vanilla. For Clear Sky and CoP though, the Complete mod is inadvisable.
I know you said you're not really into strategy, and I'm not too big on it either if I'm honest, but I'd feel remiss if I didn't say I thought Command and Conquer was a must play. Particularly Tiberian Sun through 3/Kane's Wrath. Because everyone knows that's the last one that existed. I find the very early ones tedious to get through, but I played them once. But you get them anyway if you pick up The First Decade pack, and then all you need is 3/Kane's Wrath.
Also in my library that is certainly worth playing is Bioshock, though I can't really call them MUST plays. Good gameplay and environment, not fantastic story. Infinite's was actually pretty annoying and nonsensical. I again got the feeling that Infinite wasn't really in the same series as 1 and 2.