"To go through something" means having a hard time, feeling really down emotionally.
Example 1: "Their familly went through a big financial loss." The familly struggled a lot because of the big financial loss.
Example 2: "She goes through a lot, for achieving her goals." She has a lot of difficulties on her way to her goals.
"To go through with something" means doing something, that you have planed or agreed to do, even if it's really hard. To carry through, to maintain.
Example 1: "Nobody believed that he went through with that horrible fight." The fight was really rough, but he handled it.
Example 2: "The country went through with a big crisis for couple of years." Again, the country could handle the big crisis, that lasted for couple of years.
I hope it helps!