Oh, there a bunsh out there, however it belongs to your playstyle which you should buy.
I dont know how familiar you are with all the mainstream FPS games, however i might just give you some hints here ^^
The pretty new release "Battlefield 1" itself is a very well made game. The Singleplayer was much fun and well designed. Many people like the multiplayer too, however, there are so many lemme say not that smart people out there, so I raged here and there about my team, so you should definitely find somebody to play this game with
Call of Duty has been a good fast paced infantery only FPS game, however, all last releases have been just garbage, so if you are looking for that kinda game, wait what the year brings, maybe the next CoD release wont be a flop, although i cant believe that in the moment. Try to pick up Black Ops 2 somewhere if you cant wait. There arent that many players there anymore, however, everytime i search for atleast gamemodes like Team Deathmatch or Domination i still can usually find a lobby.
And now comes the real deal. If you wanna get good at FPS games, you wanna get a challenge and have fun at getting better at a game - not simply the flashing bashing level ups and achievements - then try Counter Strike: Global Offensive. Its the cheapest option and has the best outcome. The gameplay is a little bit different and a little bit hard to learn, however there are plenty of tutorials everywhere for this game and playing it competitive is just so much fun, especially with friends and the new lobbysystem which allows you to find partys with people from your country.
This is a summary i can give about those "mainstream" FPS Games. Different Games for different playstyles. You can also take a look at ARMA which should be a pretty realistic and slow paced game, but i never played it, so i cant say anything about that ^^