I don't know if there are any "hard" rules about this is AmE. It probably depends on the area one is in. Several things come to mind in this context:
I'm going to Joe's cafe for lunch.
I'm going to Joe's cafe to eat lunch.
I'm going to Joe's cafe to have lunch.
And the once trendy, I'm going to Joe's cafe to do lunch.
When I read those sentences, I have feelings like:
The first one, is like an answer for kinda question: "Why do you go to Joe's cafe?".
The second one, seems like a task: "11 a.m go to Joe's cafe to eat lunch."
The third one, a simple sentence to say to someone, maybe your friend, what you are going to do: "Hey, I'm going to Joe's cafe to have lunch. Bye!"
And the last one, totally weird for me but, I think using this when you
have to eat lunch, you have to do it always. LOL. This one should say with a tired voice, or angry also.
Well, please leave me comments. What do you think?