YOU SAID:
the dog ate its food and then went outside.
INTO JAPANESE
犬は食べ物を食べて外に出ました。
BACK INTO ENGLISH
The dog went out to eat food.
INTO JAPANESE
犬は食べ物を食べに出かけました。
BACK INTO ENGLISH
The dog went out to eat food.
That didn't even make that much sense in English.