I'm not much of an FPS player but since the more recent FPS don't make me seasick like the old Duke Nukem and the original Shadow Warrior I've bought quite a few and I have to say I like some of these games a lot.
The only major drawback is that now I've come to realize that Fallout 3 and Fallout New Vegas have some rather shoddy shooting mechanics and that has certainly dampened my enthusiasm regarding New Vegas which I still play from time to time.
Starting with the best, Wolfenstein the New Order is every bit as fun as Rage and much more polished (but Rage is less linear) in both cases the actual shooting is amazing. I've read that people complained about the lack of multiplayer but in my opinion it's better to have a really good single player experience rather than having to rely on servers that are more often than not inhabited by campers, cheaters and obnoxious teens who (unknowingly) talk like Beavis and Butt-head.
Shadow Warrior and Hard Reset are incredibly good as well if a bit different. Shadow Warrior has the fantastic sword fighting and all the cool moves while Hard Reset is more about environmental damage and the old school gameplay (Rise of the Triad feels old school too but I don't get the same vibe from that game and the experience is less exhilarating for me).
Painkiller should be a game I would enjoy but I didn't really get into it. I've had more fun with Necrovision (first world war games are too rare unfortunately).
The Metro games are worth mentioning in this post. I haven't played the redux version (not a big fan of purchasing the same game more than once) but I've found these games to be rather fun although the linearity can be annoying but then that's what Stalker is for.
It's hard to beat Bulletstorm when it comes to senseless unabashed fun. The slaughter tends to get old but it's a real lark while it lasts. The Borderlands games have some weird humour as well although I feel the combat is slightly less solid but the open world makes up for that somehow still I don't know if I would put them in the same category (although they are good time wasters).
I could add to this list Alien Rage Unlimited since it's actually pretty nice although not in the same league (and a bit more arcade but it has a demo on steam which is becoming too rare these days).
I feel obligated to mention Far Cry 3 and Far Cry Blood Dragon since they are the best I've played when it comes to open FPS games. As much as I enjoy some of the other games I've listed, there is something refreshing about a game that takes to the open like Far Cry after being cramped in corridor shooters for too long. I haven't looked into Far Cry 4 yet but it looks like an improvement over 3 (although the island setting is more appealing to me).
A bit more arcade but still absolutely gorgeous, Call of Juarez Gunslinger is another game that is worth looking into if you like the setting. It has a great story and some infuriating bosses.
As far as stories are concerned I feel I have to mention the Darkness 2. It's the FPS in which I felt the story was more about calling forth an emotional response rather than providing an excuse for shooting bad guys in the head. It has the art and the powers make things quite refreshing.
I should mention the Bioshock series since it has powers and some pretty in depth background as well but I feel that the actual shooting in Bioshock Infinite is not as satisfying as in some of the other games I've listed above. Maybe it's the limited number of weapons you can carry or the aiming.
I haven't mentioned many military shooters because I don't care much for realistic shooters like Battlefield and I haven't touched a Call of Duty game in years (since World at War and Modern Warfare in fact). or a Halo game for that matter.
The closest I've come lately (except Wolfenstein) would be Dino D-Day but you have to agree that putting dinosaurs in a WW2 setting is a crazy idea that on paper sounds pretty enticing.
I did pick up Legendary during a Steam sale and it wasn't that bad (I enjoyed the story mostly) but these days it probably looks more like Half Life (which still makes me a bit sick). In comparison Aliens vs Predator is absolutely gorgeous (and it came as a surprise to me) but I feel you have to be a fan of the franchise to get the most out of it (but that's just my impression so far).
Last but not least I should conclude with E.Y.E: Divine Cybermancy which tends to prove that graphics aren't everything as long as you have vision and art direction. It's something special but it's not for everyone. It's unforgiving and old school in many ways.
This post is becoming a bit too long and I'm leaving out other games that I could mention like Deadfall Adventures (with a focus on puzzles), the Zeno Clash games (first person brawlers), Dead Island (not really a shooter either) or Consortium (not a pure FPS either).
I've skipped games like Aliens Colonial Marines or that Duke Nukem Forever game but I probably don't have to explain my reasons (although it would seem Aliens Colonial Marines got a patch that fixed some of the most glaring issues).
"Growing old is inevitable but growing up is optional."