Welcome to the forums!
From what you've said, I highly recommend that you check out our organized pvp events!
Tournaments are scheduled on a weekly basis, happening each Saturday.
For more casual play, there's
pvp leagues, which is organized into three month long seasons, with 3 in a year. Season 2-2016 just ended, so the third one will be popping up soon.
There's also other events, but those don't typically happen at the same time as
War, which has just entered the second round, so you'll have to wait a bit on that.