Go through can mean to experience something and finish it. "He went through a divorce." "He went through training." It also means to use something up. "He went through five printer cartridges to print the report." "He went through a case of beer. "Get through" also means to experience something, but with the added image of finishing something that was at least somewhat hard or difficult. "He got through the divorce." = "He went through the divorce and it was hard for him, emotionally. "I got through mid-terms ok" = Mid-terms required a lot of effort, but I finished them all right.

When you get through something, you successfully manage to overcome it to achieve your goal. If you go through something, it's an experience you've had. "I got through the horrible pain and went back to work" "I went through horrible pain after my car accident, and still haven't recovered completely"