Perhaps I can observe that I think there are close equivalents in English to each of your sentences (although I don't speak Japanese, sorry)
"prendere", as Irene says, "Shall we go and get an ice cream?" or "Shall we go and have some ice cream?" "Prendere", "get" and "have" are general words comprising the whole process of ordering it, paying for it and eating it.
"Perchè non andiamo a mangiare un gelato?" - "Why don't we go and eat some ice cream?" comprises literally only the eating. It sounds as if you want to eat it without ordering it or paying for it.