Translation:We are going to go to the restaurant to eat.
cummings24... "au" is short for "à le" so there is a definite article. "to a restaurant" would be "à un restaurant", but it sounds all wrong. I have heard my French friends say "au restaurant" when they meant "a restaurant", so while it is not correct, I think that is how it is used.
Yes, "pour" in front of an infinitive corresponds to "in order to". Zoevee, the use of "à" and "de" that you are referring to has more to do with the way two verbs are often linked, and the choice of the preposition (or sometimes no preposition) is determined by the first verb. For example, "I am beginning to eat lunch" is "Je commence à déjeuner" because "to begin to [do something]" is "commencer à [faire quelque-chose]" On the other hand, "I am stopping eating lunch" is "Je cesse de déjeuner" because "to stop [doing something]" is "cesser de [faire quelque-chose]. These are idiomatic, and you just have to learn them as you go along.