"It's time for dinner. So I'm going for dinner." This is somewhat idiomatic if you are talking about having dinner.
"What's for dinner?"
"Would you like to come over for dinner?"
"What do you have for dinner?"
The other sentence you mentioned, "I'm going for a dinner." It could be a contrastive emphasis such as "I'm going for a dinner, not for a party."
"I'm going for a dinner party!"
"What do you have on a dinner menu?"
"Jessica's just a dinner date."