They can all mean "first" but also have their own connotation.
1) 처음: (noun/adverb) literally means "first time / first occasion".
* 처음에: first, 처음으로(for the first time), 처음부터(from the start)
* Emphasis is on "the first stage" of something, not comparison with others
2) 일단: (adverb) as a first step. Once something happens/occurs...
* stresses "doing/acting/happening without thinking/preparing too much".
3) 먼저: (adverb) before anyone / anything else.
* implies "before something/someone else"
Examples can show their different usages.
- 난 처음에 네가 남자인 줄 알았어: In the beginning(at first sight) I thought you were a man.
- 이 일은 처음부터 잘못됐어: This matter went wrong from the beginning.
- 처음 느낌 그대로..: Just the way we felt in the beginning..
- 처음 본 사람: a man I saw for the first time.
- 일단 점심부터 먹자: Let's first have lunch (and then do other things).
- 그는 일단 시작하면 멈추지 않는자: Once he gets started, he won't stop.
- 너 먼저 가: You leave first (before I do).
- 닭이 먼저냐 달걀이 먼저냐: Which came first, the chicken or the egg.
- 먼저 할 일, 나중(에) 할 일: Things to do first, things to do later.