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.