for breakfast指的是为了早餐而去做某某事。

比如:What do you like to eat for breakfast?你早餐喜欢吃什么?

He would like to join us for breakfast.他想和我们一起吃早饭。

I eat a piece of bread for breakfast.我为了早餐吃了一片面包。

at breakfast指的是在吃早饭。

比如：listen to music at breakfast早餐时听音乐