I'm sorry but you can't use "다음에" when you wanna say in the past tense. "다음에" can be spoken when you talk about only the future. Instead, you can say "그 다음에" while you're talking about the past. (학교에 갔어요. 그 다음에 식당에서 친구를 만났어요. is right.) = I went to school, after that, I met a friend at the cafeteria. // after:다음에 + that:그 = after that:그 다음에 If you say "다음에" without "그", this meaning is closer to "later" Easiest way to make that phrase you wanted to say is "그리고", which is same as "and" in English. (학교에 갔어요. 그리고 식당에서 친구를 만났어요. is right.) = I went to school and met a friend at the cafeteria. // When you speak English you can use "and" in any tenses. Likely, "그리고" can be the easiest way to say what you want. "그리고 나서" can be spoken in past, present and future tense as well. (학교에 갔어요. 그리고 나서 식당에서 친구를 만났어요. is right.) = I went to school, and then I met a friend at the cafeteria. *"다음" is different from "다음에". "다음" is a noun and adjective, and "다음에" is used as an adverb. So, these two words can't be alternative words for each other. "다음에" can be traslated as only "later" if there is any letter before it, such as "이 다음에", "그 다음에". And the horrible thing is "다음에" is a grammatically wrong word in fact. But many Korean native use that word, so you don't need to mind using it. And "다음" is used as a same meaning of "다음에" sometimes. However this is not right. This is a kind of slang, which is short form for a word "다음에". Although I think you might think I'm too strict about this, you will make mistake that doesn't make sense someday if you learn something wrong. Ex) 다음 갈께 doesn't make sense, but 다음에 갈께 is right expression. Likely you'll find many cases, which show us small differences making big mistake.

Yes, you can use 다음에 in that context. "학교에 갔어요. 다음에 식당에서 친구를 만났어요" is okay, although it might sound a little stiff to native speakers. When 다음 is used at the beginning of a sentence as in the above, you need 에. If it's in the middle of a sentence, you can omit it: "밥을 먹은 다음(에) 밖에 나갔다", "강의가 끝난 다음 교수님을 찾아갔다", ... Also, 다음(에) is not the only or even the best word you can use in such contexts. For example: - 밥을 먹고는 밖에 나갔다 / 밥을 먹고 나서 밖에 나갔다. - 강의가 끝나고 나서 교수님을 찾아갔다. - 학교에 가서는 식당에서 친구를 만났어요. These other versions seem to be more natural than the ones using 다음에. Even in English, you don't use "next" so much but connect things with "and", "then", "and then", etc.