Effective breathing for efficient cycling It seems silly to say I don't know how to breathe, but I don't.  When doing weights, I know you exhale when you do the work part of the rep.  In Yoga similarly breathe out on the stretch.  Rock climbing too, when you pull yourself up breathe out.  The list goes on.  What on earth do I do for cycling.  

Is it different for riding on flats or going up hill?

I am talking here about when a reasonable amount of effort is being put in, not necessarily racing or time trial but not exactly bumbling along either.  I remember when I was younger and did Cross country running at school we were taught to do two short breaths in and two out.  Does that apply here or is there something else.  I saw watching time trials no TV that some of the riders seem to ride with their mouth wide open.  I tried this and found that on hot dry days it really hurt my throat and lungs.
There are going to be dozens of opinions on this, so keep in mind that I'm only talking in generalities:

Don't get too caught up on what type of terrain you're on, flat, uphill, downhill, etc. As long as you're exerting yourself on your bike, try to take deep, steady breaths, breaths that seem to go clear to your stomach. Try to focus on keeping your breath steady whenever you can. Try a few different rhythms and see if you can pick one that comes most naturally. Even better, see if you can find a riding buddy so that you two can exchange notes and learn from each other.

There are people who make their career out of training people how to breathe, so, this is just to get you started.