First and last
Positions in a sequence, going ahead of others, and first-ever experiences. These are the words for explaining steps, taking turns and telling stories.
mazu
first (of all)
まず
Opens a series of steps or names the top priority (mazu wa ...). Later steps follow with tsugi ni and saigo ni.
mazu kore tabeyou.
Let's eat this first.
まずこれ食べよう。
mazu kore o tabemashou.
Let's eat this first.
まずこれを食べましょう。
It can also mean 'almost certainly' (mazu daijoubu, mazu muri). Don't confuse it with mazui (bad, uh-oh): mazu kore = this first; mazui na = uh-oh.
saisho
first, the beginning
最初 さいしょ
Names the start of something. saisho ni = first, before anything else; saisho wa = at first, in contrast to later.
kore, saisho wa kirai datta.
I didn't like this at first.
これ、最初は嫌いだった。
kore, saisho wa kirai deshita.
I didn't like this at first.
これ、最初は嫌いでした。
hajime (ni / wa) is a softer synonym.
saigo
last, the end
最後 さいご
Names the end of a sequence or the final one: saigo ni = lastly, saigo no = the last (saigo no hi).
saigo ni kore tabeyou.
Let's have this one last.
最後にこれ食べよう。
saigo ni kore o tabemashou.
Let's have this one last.
最後にこれを食べましょう。
'Finally' after a long wait is yatto, not saigo ni.
hajimete
for the first time
初めて はじめて
Marks a first-ever experience (hajimete kita = my first time here). It is the root of hajimemashite.
kore hajimete tabeta.
This is my first time eating this.
これ初めて食べた。
kore, hajimete tabemashita.
This is my first time eating this.
これ、初めて食べました。
It does not mean first in an order. For 'first, do X' use mazu or saisho ni. nihon wa hajimete? / nihon wa hajimete desu ka? (first time in Japan?) is a stock small-talk question.
saki ni
first, ahead (of others); beforehand
先に さきに
Puts one person or action ahead of another (saki ni itte = go on ahead; saki ni shukudai yaru = I'll do my homework first) or says something is done in advance (saki ni harau = pay up front). Casual speech often drops the ni.
saki ni tabete ii?
Mind if I go ahead and start eating?
先に食べていい？
saki ni tabete mo ii desu ka?
Is it all right if I go ahead and start eating?
先に食べてもいいですか？
Don't mix it up with sakki ("Earlier and lately", a moment ago): only the double k differs. saki ni kaetta = went home before the others; sakki kaetta = went home a moment ago. 'Before him' is kare yori saki ni, since kare no mae ni usually means 'in front of him'. saki ni itte yo (先に言ってよ) = you should've said so sooner! It sounds the same as saki ni itte (先に行って, go on ahead), so context decides. osaki ni douzo = after you; osaki ni shitsurei shimasu = the set line for leaving work before others.