这个要看在句子中作什么成分,如果作主语,my father and I 正确,如：
My father and I had dinner at a restaurant yesterday.我和爸爸昨天在饭店吃饭了.
如果作宾语,就要用my father and me了,如：
Uncle Wang invited my father and me to dinner yesterday.昨天王叔叔请我和爸爸吃饭了.