GO ON A PICNIC OR GO FOR A PICNIC?

If you are talking about a place and the activity, you would probably say, "He went ON a picnic." 

The last time I went on a picnic, I brought delicious sandwiches. Yesterday, we went on a picnic.

 You might also say, he went TO a picnic if you think of the picnic like a party; you are going to an event that someone or some organization has organized.

On Mother's Day we went to a picnic at Mission Bay. The clown had a parachute we could play in.  We went to a picnic that Steve had told me about given by a Community Group. It was a rather pretty town, and a nice drive (several hours from Conway). 

"Went for a picnic" also exists. carries the meaning of "went for the purpose of," or "went in order to have".

  Yesterday, by way of a change, we went for a picnic to the shores of the Baltic, ice-bound at this season, and utterly desolate at our nearest point. 
After lessons, we went for a picnic at a beach. It was pretty. We saw surfing. I go surfing tomorrow. Day 8: Today it was cold and I went to learn to surf. 

 "Go ON a picnic," is the most usual, and it means just to have a picnic. This might be a small family picnic. "Go to a picnic" means that you are going to an event, maybe a large, organized family picnic. "Go for a picnic" means that you are going for the purpose of having a picnic.