Hereos of the Storm is pretty good to get into now that Season 1 has started. I enjoy it and a few LoL players have preferred it. (It's free)
Counter Strike Global Offensive is probably a great choice. You can get it cheap on G2A.
Overwatch is really fun but costs a lot.
Considering the steam sale coming up soon, a few singleplayer games you might like-
Middle Earth: Shadow of Mordor
Undertale
Witcher series
Metro 2033
Dark Souls series
The Binding of Isaac: Rebirth