I'm not even out of Candlekeep and two of my weapons have already broken! Once was against tiny little rats! Are weapons suppose to break often or was I just incredibly unlucky?

Because of the iron crisis normal weapons will break from time to time. Magical weapons won't break.

I believe that normal metal weapons (AKA not clubs, bows, crossbows, slings, quarterstaves) have a 1% chance to break on each hit.

Eventually, your entire party will have +1 weapons or better, and those will not break.

Whenever one of your characters roll a critical failure (roll a 1) with an iron weapon (swords, axes, etc.) they break. As strixo said, once you get magic weapons you won't have to worry about it anymore.