First thing that came to mind is Counter Strike: Global Offensive (CS:GO)
Counter Strike is a classic game with relatively simple controls/objectives. Very popular game.
Counter Strike has the worse learning curve for beginners.
I would say Call of Duty would be the most accessible to beginners. If you actually want to play against people you're going have to play the newer games like Ghosts or Advanced Warfare. If you want to get your feet wet you can try playing through the Modern Warfare series since I think those were the best play-mechanics-wise.
Battlefield 3 and 4 single player campaigns were fairly easy and can get you into the genre. Also, it you find yourself needing some practice but keep getting killed in single player games, you can use a trainer and experience the game and learn how it plays without constantly dying. I get all the trainers I use from megagames.com.