Okay, so switching from LoL, if you want a similar sorta thing your choices are basically just HoTS and Dota 2.
Overwatch is looking fine. TF2 is getting competitive mode. CS:GO might be your thing, though it literally uses the same queue as League does now. Street Fighter V?
Dotabuff has a fancy guide for switching from LoL. I can't find anything regarding HoTS, probably because it's newer.
If you're going to go FPS, do yourself a favor and don't ever fucking touch CS:GO. They took CS, stripped out pretty much everything that made it good, turned it into the closest they could get to a CoD clone, and then slapped on a fucking terrible matchmaking/ranked system while simultaneously neutering the traditional server-based system. That last has a number of problems associated with it: virtually all of the play is in shitty matchmade official servers (which don't develop the continuity or community you used to get), meaning that all gameplay takes place according to the shitty default game modes. No zmod, zescape, jailbreak, &c. and GunGame got turned into a fucking terrible mode that misses the point entirely. No server-specific tweaks, meaning no 24/7 servers, no custom maps, no non-standard player counts, an utter lack of no-AWP/auto servers, &c.
And that's on top of them fucking up the gunplay. They added in RNG; first shots are no longer perfectly accurate, and it feeds into the shitty new recoil system in which they made every gun have basically the same recoil pattern. So instead of single shot tapping and deep knowledge of how individual weapons handle, you have the same identical spray-and-pray with everything--might as well, since your shots can miss because "lol RNG" and there's barely any difference between an AK and a P90.
If you're looking to get into FPS, do yourself a favor and get CS 1.6 or CS:S. Source gets talked down by 1.6 vets, but apart from the larger hitboxes it's mostly the same experience. Telling someone to play GO is almost as bad as telling someone who wants to play survival shooters to get WarZ.