my iphone 5 charger is broken. can i go to my local apple store to get a new one for free?

Perhaps, depends how it broke. If it was due to a manufacturers defect and you are still under warranty, it would be replaced for free. If you broke it accidentally or by some sort of mis-use, then no, your warranty does not cover that. If your warranty is over, then you will have to pay for a new one regardless.

A new one costs $19. And it's a 12W charger that will work for either iPad or iPhone.

Do you have other i-devices? The chargers are interchangeable.