I can't remember crying through Buffy season 1 and 2, although I did find some scenes really upsetting.
In Season 1, Prophecy Girl, I thought the scene in the library when Buffy has just found out the she is going to die and she is arguing with Giles and Angel is very touching.
In Season 2, Innocence, when Buffy is lying on her bed crying, after she has found out that she is responsible for turning Angel into Angelus is so sad.
In the episode I Only Have Eyes For You, at the end, when Giles and Buffy are talking about forgiveness, I thought it was a great scene because Buffy looked so lost and upset, and it spoke volumes about how much Angel meant to her.
And of course, I nearly cried when Angel got his soul back and Buffy told him to close his eyes. That scene was really upsetting.