I need to go buy food is definitely acceptable in casual, spoken North American English. I am not sure about British English, though. Such a statement often includes and, for example I need to go and buy food, in the same manner as Please try and come does.
Bill's answer is great, except that I am going to buy food does not have the exact same nuance as I need to go (and) buy food: the former shows intention, while the latter shows necessity.
As far as a more formal, grammatically-correct version of the latter, I like Bill's example sentences using must and have to.