Neither of your sentences is right. The first is one sounds wrong - we'd say 'I went FOR a chicken one' in that case. The second makes no sense, because it has no object - went with what? We need to know what the options are and which option you chose. We use 'go with' - or more commonly, 'go for' - when there's a clear choice between two or more options. You sometimes hear 'go with' when game-show contestants talk us through their reasoning. For example: Compere: So what's the right answer - A, B, C or D? Contestant: Well, I know it isn't D, and I don't think it's C, either. And I've never heard of A, so....um... I'm going to go with B. Compere: B? Are you sure? Contestant? Not really, but ... But, as Jimmy says, it's much more usual to say 'go for'. The game-show contestant could equally well have said "I'm going to go for B" and 'go for' can be used in many more contexts than 'go with'. I'd forget about 'go with', if I were you.

I went for the fried chicken one. You can use 'go for' to describe the outcome of a choice like this. You could use 'go with' to describe the deciding factor, 'I went with chicken'. Similarly, about the weekend, we went with the idea of a big hotel, so we went for The Grand when we went to Brighton.

Go with somebody, isn't it?