There are random matches available at all times online. Check the person on the right side, upstairs of the Pokemon Centre. You can play ranked, or not. It's no hassle at all to do this.
There is a competitive online scene with simulators as well. It can save you plenty of time to try out a team too, even if you plan to breed all the Pokemon later. Singles is very popular in simulators, but doubles and VGC is gaining a large following.
Nintendo is also hosting online tournaments a few times per year. There's one coming up, and registration just opened for it. Sign up for an account (linking it with your game) and head over to http://en.pokemon-gl.com/gbu/ to sign up.
Awesome, thanks for the advice! I've just started the game again so I reckon it'll be a while till I'm at the point where I can play in proper tournaments.
As an aside, last night I was starting to think of a team using Smogon University tiers. Is this the standard tiers used in Nintendo and online events as well, or in a random online match are you still likely to fight someone sporting 'ubers' no matter what?
Cheers!